ASSESSMENT FOR STUDENTS' CONCEPTUAL READINESS AND
MULTIPLICATIVE REASONING
by
ERICKA J. LEONARD
B.S., Colorado State University, 1992
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science in Education in Mathematics Education
Mathematics Education
2013
This thesis for the Master of Science in Education in Math Education degree by
Ericka J. Legnard
has been approved for the
Mathematics Education Department
by
Ron Tzur, Advisor
Heather Johnson
7/2/2013
ii
Legnard, Ericka J. (M.S.Ed., Mathematics Education)
Assessment for Students' Conceptual Readiness and Multiplicative Reasoning
Thesis directed by Professor Ron Tzur.
ABSTRACT
This research thesis addressed the problem of how teachers can assess students'
readiness to begin learning and progress through multiplicative reasoning. Addressing
this problem is important, particularly because this t5^e of reasoning serves as a
conceptual foundation for higher-level mathematical topics, such as fractions, ratios,
proportions, and algebra. When students and teachers struggle with these higher-level
topics, assessing conceptual prerequisites plays a key role in identifying and eliminating
the root causes of difficulty. This constructivist premise, of the need to assess students'
current knowledge as a basis for implementing instruction adaptive to the students, is
considered essential for effective teaching and underlies this thesis study. Accordingly, a
first aim of this study was to determine the reasoning students need, both additive and
multiplicative, to learn the aforementioned higher-level concepts. A second aim was to
create an assessment instrument for teachers and researchers that would provide essential
information about students' preparedness to engage in further, meaningful study of
mathematics. The researcher used studies on students' thinking and learning to create
assessment items that would either bring to light their available reasoning or reveal its
absence. Additionally, a theory that describes the stages of student understanding in
developing multiplicative reasoning was used to better place concepts in a framework
that depicts current knowledge, and what is needed before instruction can fruitfully move
ahead. The assessment items were presented in a format and in situations that should be
ii
accessible to elementary and middle grades students and teachers. An instrument
consisting of these items was then developed to serve assessors of students' multiplicative
reasoning. This instrument includes differentiating items to fit with various grade levels
and to determine the extent to which students can solve these tasks independently or with
assistive prompting.
The form and content of this abstract are approved. I recommend its publication.
Approved: Ron Tzur
iii
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION 1
II. REVIEW OF LITERATURE 5
Constructivist Theory of Learning 5
Construction of Schemes 6
Fractions as Problematic for Students and Teachers 9
Practices for Developing Multiplicative Understanding of Fractions 10
Multiplicative Reasoning 16
Schemes for Reasoning Multiplicatively 19
Activity Effect Relationship 20
Anticipatory and Participatory Stages 23
Fine-Grained Assessment 26
Previous Research on Assessment of Multiplicative Reasoning 27
III. METHODS 30
Development of Tasks 31
Composite Unit Iteration Task 32
Additive Tasks 33
Multiplicative Tasks 37
Task Refinement 38
Construction of Prompts 39
Selection of Administration Method 43
Development of Administration Guide 46
iv
Scoring Considerations 46
IV. RESULTS 48
Tasks 48
Administration Protocol 53
Administration Guide 62
Scoring Guide 74
Reliability and Validity 82
V. DISCUSSION 84
Contributions to Research and Practice 84
Limitations 87
Implications for Further Research 89
Concluding Remarks 90
REFERENCES 92
APPENDIX 97
A. Feedback Form for Assessment 97
V
CHAPTER I
INTRODUCTION
This research thesis addressed the problem of how students' conceptual readiness
for and progression through reasoning multiplicatively can be assessed. The research
problem grew out of an interest in the issue of student conceptual preparedness for
learning fractions. The teaching and learning of concepts such as fraction and ratio seem
quite ineffective in many American school settings (Fan and Zhou, 2006). To learn
fractions, many researchers believe that students need to have previously constructed
ways of reasoning multiplicatively (Confrey and Smith, 1995). It may be that some
students are not prepared with the necessary foundation, namely multiplicative reasoning,
for instruction in more advanced concepts, such as fractions, at the time that the
instruction is delivered. Tzur's (2004) work indicated that one of the crucial steps a
teacher can take to best serve his/her students is to assess the underlying reasoning
necessary for students to build a concept before attempting to teach that concept.
Assessment of students' prior knowledge is considered important before instruction in
any case, therefore, if we consider multiplicative reasoning as an important basis for
fraction, assessing students' conceptual preparedness for multiplicative reasoning seems
critical for enabling their progress through the study of fractions.
Weak foundational skills can hinder a student for years to come (Confrey and
Smith, 1995). In particular, a lack of sufficient prerequisite knowledge may cause
students to learn the rules and procedures for concepts, such as fraction operations,
without being able to take in the fundamentals of fraction concepts. This may allow the
student to be somewhat successful in the short term, but as time goes on there can be
1
frustration in trying to understand the concepts that come later. The need for a rich,
thorough learning of fractions is clear when we consider the wealth of future
mathematical topics that students will encounter that require an understanding of
multiplicative reasoning and fractions (Thompson and Saldanha, 2003). Ratio and
proportion follow soon after fraction instruction, but the concepts of similarity, slope,
exponential and logarithmic functions, just to name a few, also draw on multiplicative
schemes. Confrey and Smith (1995) fraced such schemes extending through calculus and
beyond, and noted that they should be based on multiplicative concepts instead of
additive ones. Teachers often lament that infroducing an algebra problem including
fractions seems to dissuade many students, even if using algebra with whole numbers
seems solid for those students . This indicates that there may be a problem in the way
that students learn the foundational knowledge for fractions, and that there is an urgent
need to correct this problem.
Students who are able to efficiently and independently solve problems involving
multiplicative situations may be more able to create conceptions that support the learning
of future concepts, and those who do not can find these future topics increasingly difficult
to understand (Confrey and Smith, 1995). It seems an important issue that students are
prepared with a solid conception of multiplicative reasoning, leading to a solid
conception of fraction, and provided a clear path to future studies such as algebra.
In the context of a regular classroom experience, it is difficult for educators to
correctly determine the reasoning used as students solve problems. There are factors,
discussed in this section, which make this especially true when the assessment is taking
place mainly with pencil and paper tests, or when it is done informally through brief
2
conversations with individuals or groups of students. This is not to say that these means
of formative assessment are not valid. Rather, that when teachers need to know that
students are prepared to undertake what seems to be such important new learning, there is
a need to get an authentic, detailed picture of students' readiness.
There are many factors that make the assessment of reasoning difficult in the
classroom. These include the time-consuming need for careful selection of the problems
to be used, unintentional prompting within the problem or by the teacher or other
students, and the ways that students can demonstrate what seems to be evidence that they
understand, when in fact they may not. The ability to remove as many of these factors as
possible and to assess a student in a way that is likely to give a clear picture of each
student's capabilities is important.
As a middle school teacher, I have come to recognize that teachers need to know
the importance of this multiplicative foundation, how to detect it, and ways to support
students at their current level. A user-fi-iendly assessment of multiplicative reasoning,
which gives specific information about what students do and do not understand, would be
a tool to help teachers pinpoint where their students may need more experience before
fraction instruction begins. It is proposed in this thesis that having these assessment
results is important for teachers and students. This is not a simple matter, however,
because students who appear to have multiplicative reasoning may not have it. A reliable
assessment that helps clarify the depth of students' understanding could be valuable to
ensure their conceptual preparedness.
The instrument developed as part of this thesis study, the Assessment for
Multiplicative Reasoning (AMR), was designed to allow assessors (e.g., teachers) to
3
determine not only the student's level of multiplicative reasoning, but also whether a
student is able to reason independently or with support. The results of the AMR allow a
teacher to determine not only students' preparedness for fractions but also whether what
is needed is to lay the groundwork for a multiplicative scheme, building on the previous
scheme, or help to progress to the next stage within a new scheme. This is something
that as a teacher I will certainly incorporate into my practice. The knowledge and tools
that I have gained in this study, both the awareness of multiplicative reasoning as an
essential part of math education, as well as a way to assess this knowledge, will serve my
students well in the future. I feel better prepared to address the issues that too often arise
when I begin the teaching of fractions. I intend to use this assessment to determine the
readiness of students according to the schemes that they can access, as well as their level
of understanding within the scheme, to better target instruction for my students.
A discussion of the meaning of scheme in this context will follow in the Review
of Literature section in Chapter II. The next section depicts the conceptual framework
that guided this research, because some of the constructs from this framework are needed
to better delineate the problem addressed in this study.
4
CHAPTER II
REVIEW OF LITERATURE
In this chapter, I situate the research problem depicted in the previous chapter
within the larger body of research literature. I begin by summarizing the constructivist
theory of learning that underlies all other ideas in this work. Next, I discuss the
prevailing theories of best practice in fraction instruction. A discussion of the facets of
multiplicative reasoning and why it is important for fraction concepts follows, to link the
best practices to what we know about multiplicative understanding. Then, I will detail
the learning theory that I suggest using to support students as these practices are
employed, as well as an explanation of the participatory (prompt-dependent) and
anticipatory (prompt-less) stages of understanding that students would arrive at using this
theory. Finally, I will explore methods of assessment that can be used to determine a
student's knowledge, as well as the previous work on assessments specific to
multiplicative reasoning.
Constructivist Theory of Learning
This thesis used a constructivist theory of learning as a guiding framework,
because the focus of the assessment is on cognitive aspects of students' learning, namely,
reasoning multiplicatively. A constructivist theory asserts that students must construct
their own knowledge, and that it cannot be transferred to them through some other means
(Dewey, 1938; Piaget, 1985; von Glasersfeld, 1995). With this idea in place, it seems
that to guide students in the learning of new concepts, such as fraction, ratio, and the like,
teachers need to be aware of activities that help students make their own meaning and
understandings, and allow students to do the work necessary to create this new
5
knowledge. In constructivism, the student's thinking is a key goal for any lesson, and
effective lessons and teachers will help the student to do the thinking required to learn.
Piaget (1985), the originator of constructivism, argued that anj^hing that is to be
understood should be rooted in practical experience. Von Glasersfeld (1995) interpreted
this belief put forth by Piaget, Radical Constructivism, as "knowledge being built up by
the cognizing subject" (p. 51). If teachers are not able to relate multiplication, or
fractions, or ratios, to something that students already understand, and help them see how
to operate on fractional quantities using something tangible or figural that is well
understood, then it is unlikely that students will be able to create meaningful
understandings of fractions, let alone apply any of their understandings to other
situations. A further development of these constructivist principles follows in the next
section about the development of schemes, a key building block of logico-mathematical
thinking in the constructivist theory.
Construction of Schemes
One of the foundations in the constructivist theory of learning is the concept of
scheme (Piaget, 1985). A scheme is thought of as a 'miniature framework' in the mind,
which allows thoughts, experiences, and knowledge to be organized and connected in a
way that makes sense to the learner. Von Glasersfeld (1995) described the action on
schemes as a three-part system, comprised of a perceived situation, an activity, and a
result. The perceived situation is the learner's reality in the moment, which may be
something that they notice on his or her own, or their view of a situation created by a
teacher to inspire learning. The activity is something that is done to an object in that
situation, which may involve physical manipulation, such as pouring of liquid in a
6
geometric model, or a mental activity completed on an available scheme, such as
considering the effect of adding two negative integers when the addition of two positive
integers is understood. The result is what happens after the activity is completed, or what
the learner notices about these effects, and how they relate to the object or scheme. This
three-part system describes the way that assimilation and accommodation can take place
in Piaget's constructivist theory.
Schemes begin as simple things, such as the scheme for what a table might be,
and then move on to be much more complex, such as a scheme for things that are related
proportionally. They can involve the concrete or the abstract, but the fundamental idea
regarding schemes is that they are necessary for new learning to take place. New
information, such as the relative size of fractional quantities, must be taken in to the
consciousness of the student in one of two ways. The processes of assimilation and
accommodation, described below, take place to allow students to learn via transforming
their existing schemes into novel ones (Piaget, 1985).
The student assimilates new information into an existing scheme, meaning that it
relates so naturally for the student that they incorporate it into what they already know.
For instance, if students already know the concept of unit fractions such as 1/5, 1/8 and
1/6, and the unit fraction is well understood, when they encounter a new unit fraction,
such as 1/7, this should be a simple assimilation. The new idea of 1/7 can be
incorporated right into the present scheme because it fits nicely with the "rules" that the
student has constructed to understand unit fractions.
If, however, the student does not have a scheme for unit fractions, they will need
to accommodate, or make sufficient changes in an existing scheme, such as how a whole
7
number is composed by iterating the unit of 1, so that they can develop an understanding
of the unit fraction 1/7 (Piaget, 1985). Accommodation is a much more complex and
laborious task, what we usually call learning, and requires the student to alter the "rules"
which currently govern their scheme to allow the understanding of the new information,
or to create a new scheme that will house the new information. This accommodation, or
linking of the known to the unknown, is what is referred to in the future sections
describing the ways that multiplicative reasoning can be used as the scheme which will
be accommodated for a student to understand fractions.
It was further conjectured by Piaget (1985) that students relate new information to
what they already know, even if what they know is incomplete. If not properly guided,
this connection may be tenuous and without any mathematical meaning, causing the new
learning to be unstable or misdirected. Teachers see this phenomenon when students
express misconceptions or convoluted reasoning in arriving at a solution that seems to the
teacher to be inexplicable. Presumably, the student made some connections in past work,
which functioned for them at the time in some way, but perhaps only coincidentally or
tangentially. This may happen if the underlying scheme is not properly developed, such
as may be the case with multiplicative reasoning and future concepts.
In the context of this thesis, the development of a scheme as an accommodation of
previous schemes is central, because it builds the case for educators to be mindful of the
schemes that student have available, through some sort of assessment, and how we expect
them to elaborate those schemes to include new knowledge. This understanding helps to
guide the later sections, to provide a foundation for the need that drove the creation of
this assessment, and explore the theories with which it was constructed. Giving students
8
a foundation in multiplicative reasoning, and creating experiences for them that will show
fractional quantities as multiplicative relations may be a solid path towards real
understanding of concepts such as fractions, ratios, rates and proportions. I suggest in the
following sections that one of the problems that teachers face in teaching fractions is that
the multiplicative schemes needed to proceed are not fiiUy in place, and cause students to
have great difficulty in learning. Evidence for that difficulty is presented next.
Fractions as Problematic for Students and Teachers
As students are attempting to assimilate and accommodate their schemes
throughout their mathematics education, some seem to come more easily than others.
The conception of fraction is often difficult, as the studies in this section will show.
The familiar International Mathematics Studies, including SIMS in 1976 and
TIMSS beginning in 1995, as well as the International Assessments of Educational
Progress (lAEPl and 2), showed US students lacking in mathematics in general, and
fractions, ratios and multiplicative reasoning, a prominent part of all of these
assessments, were no exception (Fan and Zhou, 2006). The National Mathematics
Advisory Panel stated, "The most important foundational skill not presently developed
appears to be proficiency with fractions ..." (2008, p. 1 8). It seems that this is an area
where American teachers and students need assistance and continued focus to create
improvement.
Fractions are a topic often dreaded by students and their parents, and cause
teachers from middle school through high school difficulty and frustration when students
who should have learned the concepts long ago struggle with using them to solve
problems. Students have been seen in studies to avoid operations such as multiplying by
9
fractions, even when this would be the most efficient strategy for solving a problem, and
they have had experience with these methods before (Fischbein et al., 1985). This is
presumably because they felt uncomfortable with the meaning of such an operation and
often preferred to cling to more familiar, additive methods. This may be due, in part, to
the evidence that multiplicative quantities are not intuitive in the way that additive ones
are, and that students will generally not develop these understandings without careful and
explicit instruction (Dehaene, 1997; Fischbein et al. 1985; Steffe, 2002). It has been seen
in multiple studies that in order to come to a clear understanding, students should be
assisted in their learning process, usually in school by their teachers and through social
interactions with others, in order to develop the schemes necessary (Dewey, 1938).
While there is no single universal theory for what should be done to bolster the
fraction proficiency of US students, there are many theories that intend to guide educators
in this endeavor. Many researchers have examined the teaching and learning of fractions
as they attempted to unravel its complexities. Some of the main theories are presented
next in order to understand which among them seems to be the most promising, and what
prerequisite knowledge is required in each case.
Practices for Developing Multiplicative Understanding of Fractions
If fraction education is problematic, it makes sense to explore the prevailing
research about how the teaching and learning of fractions might be approached and
understood. The theories presented here are highlighted to show the rationale for the type
of reasoning that I tried to assess using the AMR. There are many options for
infroducing fractional concepts to students, including, but not limited to, part-of-a-whole,
fractions as division operations, whole number bias, and the reorganization hypothesis
10
(Siegler et al. 201 1; Stafylidou and Vosniadou, 2004; Steffe, 2002; Thompson and
Saldanha, 2003;). As a teacher, I found exploring these theories indispensable for
understanding the different ways of thinking about how to teach fractions, and to
determine which theory seemed most likely to produce effective learning. The sections
that follow explore what types of schemes students need to develop during the study of
fractions, and then to consider the underlying schemes that would need to support that
study. From this understanding, the assessment developed to focus on these underlying
schemes.
Part-of-a-Whole Model
A prevalent method used to help students think about fractions in American
schools is the part-of-a-whole model. In this model, the student sees a whole divided into
a given number of equal-size parts, and consider the fraction as one part, or several parts,
of that greater whole. This part-of-a-whole conception breaks down, however, when
students are confronted by a quantity that does not fit this model, such as 8/5, because 8
cannot be a part of five (Thompson and Saldanha, 2003). When students are asked to
operate on these often mysterious quantities, they may be forced to rely on taught or
assumed rules, which may have little grounding in mathematical theory. This process is
understandably confusing, and may contribute to student thinking that math is a set of
arbitrary rules to be followed, regardless of intuition or prior knowledge (Erlwanger,
1973). The true understanding of mathematical concepts is a major purpose of valuable
mathematics education (NCTM, 2000), and when students are not able to make sufficient
meaning, not only does their understanding break down, but their enthusiasm for the
discipline as a whole may break down as well. It is considered crucial in this thesis that
11
students understand fractions as multiplicative quantities instead of as part-of-a-whole.
For example, they need to think of a unit fraction, say 1/6, as a unit such that the whole is
6 times as much.
Fractions as a Division Operation
A related understanding of fractions as multiplicative constructs, in place of or in
addition to the part of whole model, is to see them as the effects of a division operation.
In such a model, 5/8 means five divided by eight, which is a valid way to consider this
notation in certain circumstances, and which will also yield the decimal form of this
quantity. However, such a notion can be problematic when one encounters, for example,
5/ (8/3), because the operation to be completed is not clear (Thompson and Saldanha,
2003). This way of introducing fractions may also be dangerous to student understanding
because it does not provide a way to think about the value of the quantity presented, nor a
way to compare and order fractions without completing the operation and comparing the
decimal. This lack of understanding of the magnitude of fractional quantities can create
difficulty for estimation and using intuitive means to determine whether the solution to a
problem involving a fraction operation is reasonable or not. On the other hand, 5/8 could
be thought of as a unit that is 5 times as much of another (1/8), where the whole is 8
times greater than 1/8, as explained for unit fractions. Such a conception (e.g., 5 units of
1/8) supports understanding of fractions in a way that does not seem possible with the
division operation model alone.
Linking Additive and Multiplicative Situations
Another theory, proposed by the Rational Number Project (Behr, et al., 1983;
Cramer et al., 1997), is that the comparison of additive and multiplicative situations be
12
made explicit, so that students see examples of each, side by side. The theory supposes
that the students are then better able to use this knowledge to understand that fractions are
multiplicative quantities and can be treated as such. This avoids the common lack of
understanding by students, for example, that in making an equivalent fraction you may
just as easily add or subtract the same quantity to or from the numerator as multiply or
divide by the same quantity. This type of misconception may show that the student does
not see the fraction as a quantity involving a multiplicative relation.
One theory related to this stance is whole number bias, or whole number
interference (Stafylidou and Vosniadou, 2004). In this approach it is argued that when
students apply what they already understand about whole numbers to understand
fractions, confiision occurs, and that we should not try and connect the two for students.
For example, students may mistakenly believe that division by a fraction smaller than 1
results in a quantity smaller than the dividend, similarly to the result of division by a
whole number. While drawing upon prior knowledge is the only way for students to
create new knowledge in the constructivist view, it does present problems if students
make associations that are not helpful in understanding the concept. Stafylidou and
Vosniadou (2004) and Streefland (1991) stated that the conceptions students have about
natural numbers create problems for students when they encounter fractions, because they
generally draw incorrect conclusions, or make connections between ideas that are not
mathematically sound. For example, students sometimes reason that as the numbers in a
fraction get larger, the value of the fraction become larger as well (e.g., 9/10 vs. 90/100,
respectively), regardless of the invariant, multiplicative relation between numerator and
denominator. The study conducted by Stafylidou and Vosniadou (2004) asked students
13
to compare fractions and give a rationale for their conclusions. The responses were
categorized according to their conceptions. The conception called Relation between
Numerator/Denominator was the only one in which students showed a comprehensive
understanding and strong conception for all types of fraction relationships, including
improper fractions. Interestingly, even though this theory contrasted with the one that
follows, the importance of multiplicative reasoning prevailed in both cases.
The Reorganization Hypothesis .
An alternative theory to the whole number bias, and that supports the
development of fraction from whole number multiplication, is called the reorganization
hypothesis (Steffe, 2002). It states that the number system that children learn in early
grades, should be expanded, or reorganized, to include all rational numbers instead of
separating whole number and fractional quantities for students (Siegler et al. 201 1; Steffe,
2002). This theory finds value in the struggle to integrate fractions and whole numbers as
members of a continuous number system, and to create a comprehensive mental number
line through which students not only gain a strong understanding of the values of rational
numbers, but also a richer understanding of whole numbers. This "marriage" of whole
number and fraction is also supported by recent neurological research which indicates
that the areas of the brain in which fraction magnitude decisions are made occurs in the
same area of the brain as whole number processing, and that absolute magnitude and
relative magnitude are processed with "the same analog code" in the brain (Jacob and
Nieder, 2009, p. 4656).
Steffe 's reorganization model (2002) asks students to equipartition to find the unit
fraction, and then iterate it to create the desired fraction. As Tzur (2004) emphasized, the
14
key in such an approach is building on the child's activity part of a scheme (here,
iteration of unit, which underlies the creation of whole numbers) instead of on the child's
abstracted construct of number per se. Another aspect of Steffe's theory is to use
splitting, which asks students to coordinate both partitioning and iterating as inverse
operations. This is context-independent and applicable to many other situations,
including improper fractions, and underscores the multiplicative relationship by focusing
on the relation between the unit rate and the number of units.
Ni and Zhou (2005) also advocate a merging of teaching whole number and
fraction concepts concurrently, to emphasize and link the known whole number concepts
with the newer fraction concepts. This can allow students to accommodate their whole
number schemes effectively to understand rational numbers.
The basis for all of these strategies is multiplicative in nature, and calls for
students to have a strong background in the ways quantities can be related
multiplicatively in order to make sense of the tasks presented. It is considered for the
purpose of this thesis, and the associated instrument, that reorganization is preferable as a
method for guiding students in their understanding of fraction. This thinking, guided the
development of the assessment, discerning between schemes available to students that
can be reorganized further toward intended mathematical concepts.
It is important to note the conclusions of researchers regarding the most effective
ways for students to learn fractions, because many students, as well as adults, assume that
multiphcation is simply repeated addition. Such a conception does not seem to support
fraction understanding as effectively as a more robust conception of multiplicative
relationships. The next section focuses on ways of reasoning multiplicatively that were
15
seen to be paramount in the theory of reorganization, and how such reasoning may
support students in their future learning.
Multiplicative Reasoning
In the previous section, I have provided a rationale for multiplicative reasoning as
the conceptual basis for strong fraction understanding. In this section, I provide more
details about what that reasoning entails. The understanding of multiplicative reasoning
for whole numbers that is used in this paper was summarized by Steffe (1992) in the
following statement; "For a situation to be established as multiplicative, it is always
necessary at least to coordinate two composite units in such a way that one of the
composite units is distributed over the elements of the other." To make sense of this
foundational assertion, one must understand the idea of number as an abstract,
sjmibolized, composite unit.
A composite unit is an abstract construct, or scheme, that a person uses to
sjmibolize potential (but not actual) results of counting, such as a numeral or number
word, which means for him or her, a collection of units of ones. When a student says
"4", and is using it to refer to a unit of four ones considered as a group, the student
understands composite unit. In earlier stages of development, a student may recite the
word "four" in rote counting, or know that it goes with the numeral 4, but she must
develop the idea that the number four is a group of four ones which can be assembled and
re-assembled (Olive, 2001). For example, understanding that 5x7 is not simply 5
repeated 7 times, but five units, each composed of seven units of 1, or seven units of five
units of 1, and why they yield the same number of Is in total, is necessary for students to
understand multiplication.
16
A repeated addition model misses the coordinated units idea of multiplicative
reasoning. Several of the core ideas that help students understand multiplication in a
flexible, complete manner, such as the importance of units as a point of reference, are not
supported by repeated addition. In a repeated addition model for multiplication, we can
consider, for example, three pies made of six apples each, which total eighteen apples (or,
in repetition, 6 apples + 6 apples + 6 apples =18 apples). In a multiphcative model, we
coordinate the one composite unit of six apples per pie (unit rate) with three pies as the
number of composite units, which when distributed as explained above produce eighteen
apples as the total of Is, coordinating two different units to yield a third unit). In the
multiplicative situation, one must keep track of whether the number refers to apples,
apples per pie or pies themselves, and distribute the six apples across each of three pies to
find the result. When thinking of this situation as repeated addition, this coordination can
be completely missed by students.
The apparent weakness of repeated addition is confirmed by Ni and Zhou (2005),
who stated that not only does repeated addition prove a weak and limited view of
multiplication, but that this reasoning is specifically harmful to students in learning
fractions. One issue is that seeing 5x7 as five groups of seven is conceptual, which is
essential for building schemes and understandings, while seeing 5x7 as seven added five
times is simply a command to act. When seen as a command, an expression such as 5
times X, or 5x, becomes problematic for students, as they are not able to carry out the
specified computation (Thompson and Saldanha, 2003).
Unfortunately, multiplication as the coordination of units explained above is not
something that teachers can take for granted as understood by their students. In a study
17
by Clark and Kamii (1996), only 49% of fifth graders had strong multiplicative
reasoning, at a time when formal instruction in multiplication is likely finished in the
classroom. Furthermore, Siemon, Breed, and Virgona (2005) found that far too many
students at the secondary level were using simplistic additive models when multiplication
would have been more efficient. It was considered likely by the authors of the study that
part of the reason for their lacking proficiency was that they understood multiplication as
repeated addition. Steffe (1994) said that when students use an additive model to
multiply, it is an indication that they have yet to dissociate the idea of multiplication from
more primitive counting schemes.
The difficulties pointed to by the aforementioned studies show that there could be
benefit derived fi-om an instrument to determine where students seem to be on the
continuum from additive to multiplicative reasoning before multiplication is considered
solid enough to begin further instruction in tractions. In this sense, multiplicative
reasoning is a conceptual prerequisite for middle grades mathematics (Thompson &
Saldnha, 2003; Tzur et al., 2013). Its absence may create significant problems for
students who are not yet equipped to construct these understandings. Most importantly,
an assessment instrument is needed because teachers may not realize that this gap in
reasoning multiplicatively is present. It is often assumed that students are ready for
further instruction if they are proficient in executing calculations of multiplication and
division, and can give some reasoning that supports their work. The notion that these
students may not indeed have a full multiplicative understanding, complete with the
ability to track and anticipate the coordination of units involved in problem situations,
was paramount in the development of the AMR.
18
Another reason to develop an assessment instrument of multiplicative reasoning is
that many teachers may not be aware of the schemes and stages involved in multiplicative
understanding. This is especially true at the secondary level, in which these concepts are
not usually taught. Jacob and Willis (2003) commented on the need for teachers to be
aware of and able to distinguish between additive and multiplicative reasoning in order to
better serve their students. Seeing students "skip count," which may reflect essentially a
facility with repeated addition, can give the impression that students understand the
concept of counting groups of objects. However, if this is simply a learned method, and
not rooted in an understanding, it may be only an indication of a willing student. The
next section will focus on the specific schemes that students develop when constructing
meaning for multiplicative situations.
Schemes for Reasoning Multiplicatively
The multiplicative reasoning described above does not develop all at once, but on
a continuum of schemes. The framework for student understandings discussed in this
work, came trom the six schemes of multiplicative reasoning proposed by Tzur et al.,
(2012). These schemes begin with multiplicative Double Counting (mDC), during
which students are able to keep track of two quantities when one is distributed over
another. This first of six schemes, mDC, is the highest level assessed in the AMR
instrument, because it focuses on the shift in thinking from additive reasoning to
multiplicative reasoning for the student. For example, being able to find the total number
of flowers in four vases, if there are three flowers in each, would require a student to be
able to count how many composite units of three flowers are present, while at the same
time counting how many units of 1 single flower are accumulating respectively. They
19
would then need to knowingly stop when the count of the composite units of threes
reaches four, and understand that the number obtained is total flowers, not vases nor
flowers per vase.
This distribution is an advancement in thinking from the additive, in which a
student would either need to count each flower, or add 3 + 3 + 3 + 3 to find the result of
12, but misses the idea of the units being coordinated. The meanings of the answer and
the two numbers 3 and 4 are critical, as well as which is being distributed across the
other. Failure to understand this can become problematic in later studies if this concept is
not solidified early on. The six schemes, of which mDC is first, have laid a foundation for
educators to not only determine where students may be in relation to multiplicative
reasoning, but to ensure that understandings necessary for meaningful work on fractions
are in place.
Activity Effect Relationship
One elaboration of the constructivist learning theory was articulated by Tzur and
Simon (2004) in their depiction of the mechanism of learning, namely, reflection on
activity-effect relationship (AER). This depiction described the way that a student would
engage in an activity, which may be visible to an observer or occur mentally, to achieve a
goal or desired outcome that the learner has in mind. This activity is performed on an
object, which again may be physical or simply a prior conception (i.e., number as an
absfract, composite unit). The effect is what follows from the activity performed on the
object. It stands to reason that if there is not a viable object, the activity cannot proceed
fruitfully, and that the teacher must be aware of the mental objects available to the
student. When the object is acted upon an effect will be created, but the effect and
20
learning may not be optimal, and possibly even harmful, if it creates misunderstandings
on the part of the student due to insufficient prior conceptions.
To illustrate the working of this theory, consider a student who has a
multiplicative scheme such as mDC, as may be measured by the AMR, attempting to
solve a fraction problem, and how that student may be instructed using the model of
AER. A student with solid multiplicative reasoning could independently solve a problem
such as this one: Maria wants to have juice boxes at her birthday party for all of her
guests. The juice boxes come in packs of 4, and the store has 6 packages on the shelf
How many juice boxes will Maria have if she buys all 6 packages of four?
A student who has multiplicative reasoning could solve this problem by iterating
the composite unit of four six times to efficiently find that there are 24 juice boxes to be
purchased. When it has been established that the student is able to operate in this way,
the multiplicative scheme being used can be extended to allow the student to think about
fractions, such as considering what fraction is just one pack of all 24 items). This is
exactly the type of reasoning that the AMR can detect to determine whether a student is
ready to advance to fraction. A student may then be presented with a problem such as
this one: Maria is cleaning up from her party, and finds that some of the juice boxes she
served her friend are not finished. She finds five juice boxes that are each % full. If she
puts them all together, how many boxes could she fill completely?
Using a continuation of the previous problem can help students see the connection
between the two, and begin to activate the scheme, but they will likely need further
direction to fiiUy utilize the current scheme for this problem. The student may be
prompted to draw the boxes, and to describe what they see, redrawing the juice boxes
21
when combined. The student could then be prompted to use the same idea that was used
in the multipUcation problem to describe the total. By reflecting on the commonalities of
six units of four juice boxes, and five units of % juice boxes, a student can come to see
that the total would be 5*1/4 , or 5/4 of a juice box. This can mean for the student that it
takes four of these units to make a whole, and that five are being considered, so I will be
able to fill up one juice box with four units and will then have one left over, the same as 1
% juice boxes. If the student is comfortable with the multiplication to find the number of
juice boxes in the previous problem as 6 * 4 or 24 juice boxes, this same conception can
be used to understand fraction operations.
This transition is not likely to occur quickly or easily, but can occur with
continued focus on the conceptions that students have, and continued emphasis by the
teacher on the common patterns with iteration of whole numbers and fi^actions. By
asking the student to reflect upon their work for each type of problem and reporting on
their findings, a solid foundation for the understanding of fractions can occur. The key in
this process is for the student to reflect upon, and make sense of what they see as it
relates to what they already know. It is important that they do this reflection while being
directed toward reactivating the pre-existing schemes that will best support this new
learning, using multiplication to support an understanding of fi^action. This
understanding may begin with students needing support and prompting from the teacher,
but with assistance, students will become independent even when the situation or context
is different. The reflection on AER model helps us to understand these two stages, called
anticipatory and participatory (see below), as well as the need for fine-grained
assessment, which is described in a later section as a way to discern between these stages.
22
Anticipatory and Participatory Stages
That students may be at different stages when constructing a new scheme was
considered in the development of the assessment in this thesis, and these two stages are
explained in the section that follows. Participatory and anticipatory stages were
considered important because in order for students to be able to function effectively in
their multiplicative schemes, they would need to be at the later stage described here.
Also included in this discussion are the ways to determine the stage in which a student
seems to be functioning, a critical part of the AMR.
Tzur & Simon (2004) postulated that students may have constructed a new
scheme at a stage, called participatory, which requires prompting in order for them to
access, and use, the newly forming scheme. Later, the students may progress to a more
advanced stage, called anticipatory, in which they can independently use the new scheme.
This distinction implies that students may be able to solve a problem given
manipulatives, following a lesson on that concept, or with prompting, but they would not
be able to solve independently, or when the memory of the activity has faded. Tzur &
Lambert (201 1) related this cognitive stage to Vygotsky's notion of Zone of Proximal
Development (ZPD), suggesting that the ZPD is a reflection of students' cognitive need
for prompts.
A student who is at the participatory stage may be able to solve a problem about
the number of flowers in four vases with three flowers each, during or immediately
following a lesson in which these types of problems were posed, and worked through
with guidance from the teacher (e.g., "How about using our fingers to keep track of how
many vases and how many flowers we have in all?"). After some time, however, the
23
student may seem not to be able to solve problems of this type at all. This, sometimes
called the "next-day phenomenon", is a common stumbling block for teachers and
students, causing frustration for both parties. Possibly, the issue here is that the student
might have been only at the participatory stage, meaning that they must be prompted, at
least mentally, to be able to reason about the problem. The student seems to understand,
and may be able to solve independently after prompting, but the next day in class the
student may not remember how they should proceed on a similar problem. For example
if the teacher asks questions such as "How many flowers were there in one vase?" "Now
how many would be in two vases?" to lead the student's thinking, it may quickly result in
a student's ability to resume their previous way of reasoning and solving the problem.
The student continues to need prompting, however, and will likely not be able to solve
these tasks as time goes on and the activity becomes a distant memory.
At a higher level of the participatory stage, a student may prompt oneself upon
realizing that he made an error, sometimes called the 'oops' experience. For example,
students may begin the vase problem by saying there are seven flowers (e.g., adding
4+3), and then correct themselves as they realize that there are three flowers in each of
the four vases and answer correctly. This is still not considered anticipatory, because the
student needs to self-prompt. However, this is a higher level of the participatory stage,
because the prompt has now come from within the students' mental system (Tzur &
Lambert, 201 1). One difficulty in discerning between the levels is that unless the teacher
is very involved with the student during the solution process, the self-prompting may not
be evident. The student will correct and then move on, leading the teacher to assume
independence in using the new scheme, and thus anticipatory status.
24
Unlike at the participatory stage, a student at the anticipatory stage does not need
to consider the activity, or to be prompted. Their understanding is more solid because it
occurs in the student's mind as a well-formed scheme, independent of working through
the activity in the familiar context. In the example above, the relationship between
flowers and vases is clear to the student, and they don't hesitate to proceed, anticipating
the entire process from the start. They anticipate the need to coordinate counting of one
composite unit (4 vases) with another (3 flowers/vase) to calculate the total (e.g., 1 vase
is 3 flowers, 2-is-6, 3-is-9, 4-is-12). This anticipatory stage can become a conceptual
foundation for meaningful memorization of the fact 4 x 3 =12.
It may be that in the classroom some students do not reach the anticipatory stage
of a concept, such as multiplicative double counting, before the instruction moves
forward. For students not at the anticipatory stage a problem arises, because though they
have been exposed to a concept, and made progress toward understanding at the
participatory stage, they are not prepared to independently use that understanding as a
building block for new learning before the topic of study changes. This situation is also
frustrating for teachers, who feel as if the student should have learned the material
already, and showed signs of understanding, but cannot access this concept or scheme
when they attempt to move forward. In the teaching of concepts such as fraction and
ratio, a participatory understanding of multiplicative relationships is not sufficient to
create understanding about the meaning of fi-actions, so the detailed assessment of this
reasoning is crucial. The next section discusses the type of detailed assessment that can
shed light on these distinctions, and was used in the development of the AMR.
25
Fine-Grained Assessment
The assessment technique used in the AMR is one developed by Tzur (2007) to
better assess the precise level of student understanding, by distinguishing between the
anticipatory and participatory stages explained in the previous section. In fine-grained
assessment, it is important that the items begin without prompting, and progress to less
difficulty and more support, in the form of prompts and questioning, to determine
whether the student is able to reason independently. When students are presented with
problems, it is without any prompting or introduction by the assessor. If the student is
unable to solve the problem, prompts are then given in a way that gives the student clues
about how to act, while being cautious to give as little away as possible. The intent is to
find the highest level at which the student can function. If the student is given an easier
problem first, or is led toward that higher understanding by instructions, the reasoning
seen may be as a result of the unintentional prompting the student receives, instead of the
student being able to anticipate the entire process on her or his own.
For example, if one desires to know if a student can find a missing second
addend, the problem should be presented without any prelude. The student may be
asked, "I had three candies, and my friend gave me some more. Now I have seven
candies. How many did my friend give me?" If the student is able to provide reasoning
and solve this independently, they can be considered at the anticipatory stage. If they are
not able to solve this problem, the assessor could prompt, beginning with calling attention
to the numbers in the problem and what they mean. The assessor might ask: "How many
candies did I have to start with?" and then ask the student if they can solve. A follow up
could be: "How many candies do I have at the end?" and then: "Can you use the number
26
I have now and the number I started with to find how many my fi-iend gave me?" If the
student can use these prompts to solve, they are at the participatory stage for this
conception. They have some knowledge of how this problem can be solved, but need to
be prompted to arrive at a solution. If the student is not able to solve even after
prompting, it is likely that they have not begun to build the concept for themselves at all.
Additionally, it is important that student assessment not follow an activity in
which they were engaged in solving similar problems. This is to avoid the participatory
student appearing to be anticipatory. Immediately following a lesson, the student can
simply recall the activity, and use it to solve the problem. It may be impossible to tell
whether the student can reason this way independently or only following the lesson.
The importance of this distinction between participatory and anticipatory students
is that even though the participatory students may appear to understand the concept in the
context of the current class period, or with subtle prompting, these participatory students
will likely not be able to draw upon this reasoning to develop later concepts, and could be
left behind. This shows the importance of an assessment that can distinguish between
these stages, so that planning for these students can be optimal and directed specifically
at their area of need. The next section shows previous work in assessing multiplicative
reasoning, which was used to guide the development of the AMR, and to show the
differences between these attempts and mine.
Previous Research on Assessment of Multiplicative Reasoning
There are some existing instruments that have been developed for the assessment
of multiplicative reasoning necessary for fiiture concepts. Here I will present these
instruments, with what I consider to be their strengths, and then explain how the AMR
27
attempts to go one step further to achieve a fine-grained assessment of multiphcative
reasoning.
One instrument, developed by Siemon and Breed (2010), was a pencil and paper
test to address many of the same concerns expressed in this paper. The assessment
considers many of the same conceptions, including distributing one composite unit across
another. That instrument has excellent potential to determine whether students are
reasoning multiplicatively, and includes rich tasks for them to explore, moving beyond
what is assessed in the AMR to include proportional reasoning. The intent of the Siemon
and Breed's (2010) assessment is similar to the AMR in that it tries to capture student
thinking and elicit strategies used to determine the sophistication with which the student
is approaching the problem, and the resources that they employ to solve them. Scoring of
this assessment was rubric based, and completed after students had finished the problems.
Their instrument, however, did not include a distinction between participatory and
anticipatory stages in a child's use of the mDC scheme, and did not provide the same
feedback as a clinical interview situation could.
Another assessment, developed by Bright, Joyner and Wallis (2003), has similar
features to the Siemon and Breen instrument. This assessment included a multiple choice
paper and pencil test, but mentions interviews to follow up and question students about
their answers to delve deeper into their thinking. It was noted in their work that it is
possible that questions which are easier to answer, such as multiple choice questions,
may induce more sophisticated reasoning, while open ended questions may cause
students to revert to more primary methods which they feel most comfortable using.
Their point may be captured when using the participatory/anticipatory stage distinction.
28
for which specific, finer considerations and ordering of the tasks is needed (Tzur, 2007).
This consideration guided the development of the AMR, to allow assessors to
differentiate between anticipatory and participatory stages of mDC. The AMR attempts
to be more specific than the previous instruments in assessing the smaller increments in
reasoning. In the Methods section, the development of the AMR is detailed, using this
stage distinction as an important part of the assessment.
29
CHAPTER III
METHODS
This chapter chronicles the development the AMR, including the selection of
tasks, the administration method, the administration guide, and scoring. The Assessment
for Multiplicative Reasoning (AMR) was developed to give teachers and researchers a
way to assess the level of a student's multiplicative reasoning, or preparedness to engage
in that reasoning. The intent is that instruction can then be focused at the appropriate
level to give students the experiences that they need to develop the next level of
reasoning required to support multiplication, or to indicate that the student is ready with a
strong basis to experience other math concepts for which multiphcative reasoning is a
prerequisite, such as fractions, ratio and proportion, and slope.
The AMR was developed with assistance from experts, and then tested briefly on
a small sample of fourth grade students. The full final version of the assessment to date
with all of the components mentioned here is provided in Chapter IV, but examples are
given in this chapter to illustrate the development process. This chapter explains the
process by which the assessment grew from tasks to target specific schemes of
multiplicative reasoning, possible distractors that were taken into consideration, and the
design of prompts to bring to light the possible participatory and anticipatory stages at
each conception. Later sections describe the development of an administration guide for
assessors and considerations for scoring the results. Refinement based on feedback from
experts and from the initial testing are discussed in each section as appropriate, as well as
their implications for the improvement of the AMR.
30
Development of Tasks
Tasks were created based upon theoretical, literature-based analysis and
on my interpretation of the performance that students should exhibit if they have
completely developed the scheme in question (Zazkis and Hazzan, 1998). Through
conversations with Dr. Ron Tzur, I decided to include prior schemes in the assessment:
Counting On, Missing Addend, and Composite Unit Iteration (shown in Chapter IV), to
screen students and determine whether the additive schemes, rooted in number as
composite unit and necessary for multiplicative concepts, were accessible to the student.
One multiplicative scheme was included as well, namely Multiplicative Double Counting
(mDC), to assess the progression of students from additive to multiplicative reasoning.
The tasks were differentiated into grade bands, in an attempt to allow for
flexibility in administration by providing multiple entry points for students and assessors
to use, being responsive to previous knowledge about student understanding. For
example, a student at the fifth grade level may be tested at the 3-4 grade band, if the
teacher or assessor knows that the student will struggle with the numbers involved at their
grade level. It is also advised, and stated in the Administration Guide in Chapter IV, that
in general the assessor should begin with the grade level task, but move down to the
previous level if the student is unable to answer. The intent here is to determine whether
the struggle is due to the numbers involved, or to the conceptual scheme or stage being
assessed. The following section provides a description of the tasks used to assess these
schemes, and information about the choices made in their development.
31
Composite Unit Iteration Task
Composite unit was explained earlier, and is important for both the additive and
multiplicative schemes. The Composite Unit Iteration task asks students to use these
units and iterate them to find a new quantity, without counting all of the items. This
requires students to coordinate two quantities, the unit to be iterated and the number of
iterations that take place. This task is administered first because it involves composite
unit, which is important to both additive and multiplicative thinking, but also because it
represents a midpoint in the continuum of the assessment as a whole. This approach was
used although there was not enough data fi-om student responses to create a proper
Guttman Scale (Trochim & Donnelly, 2008) for these tasks. Such a scale requires one to
use data to create a linear ordering of tasks. However, an attempt was made to arrange the
tasks in a developmental order that would most likely occur for the student conceptually.
Thus, if a student can properly solve and reason about the first task the assessor can move
to higher levels, and if not, the assessor would go back to prior conceptions. For
instance, it is unlikely that a student who can perform Multiplicative Double Counting
would be unable to solve a Counting On problem.
The original Composite Unit Iteration Task I had created, seen in Figure 3.1, was
problematic for students. It was intended to give students a visual model and see if they
could use the groupings of four beads to iterate and find the total, but students were
generally not able to use this information to iterate correctly. It was suggested by Dr.
Tzur that I eliminate the task, and it no longer appears in the final version. Instead, a task
involving iteration of the smallest composite unit (i.e., 2) was created, with a context
32
using pairs of socks that seemed familiar to any student. This task is further described in
Chapter IV.
How many beads will you need to make this necklace as shown?
\ /
Figure 3.1. Previous Version of Composite Unit Iteration Task
Additive Tasks
The additive portion of the assessment, including the schemes of Counting On
and Missing Addend, is not intended to be an exhaustive assessment of additive
reasoning, but to give a rough idea of where a student might be flinctioning in regard to
additive schemes in general, and in terms of her use of composite units in particular.
These schemes are explained next.
Counting On refers to the process by which a student can begin with a quantity,
such as 7, and then add to that quantity, perhaps 4 more, and find the total. The student
does this not by starting to count from 1 , but by beginning with the known number 7 and
counting from there. This counting on strategy is sometimes called double counting, to
emphasize that the student is purposely keeping track of Is in the second addend and
knowing to stop after they have counted four more (Tzur & Lambert, 201 1). The student
might say something like this: "I have 7 already, and four more; so eight is 1, nine is 2,
ten is 3, and eleven is 4. I have eleven all together." The Counting On tasks in this
assessment were constructed so that the second, added quantity is smaller. This was done
to make representing these problems with figural, pictorial or manipulative strategies
more accessible. Assessment of students' abilities to reverse the quantities for counting
33
on when the larger quantity is given first was not a part of this assessment due to the
limited nature of this portion.
The next two schemes are assessed using Missing Addend problems, where a
student is given one of the quantities to be added and the total in a problem situation, and
is asked to find the other added quantity. In the first type of missing addend problems,
the student is given the first addend and the total, and is asked to find the second addend.
In the second type of missing addend problems, the student is given the second addend
and the total, and is asked to find the first addend. These problems are considered
different for students, as the Missing First Addend requires more sophisticated thinking.
The Missing First Addend Tasks are considered more difficult, because of the way that
the composite units are nested within the whole (Fuson et al., 1988). If the second
addend is missing, the student can know where to start her counting, as they do in
counting on, and proceed by counting up from the first addend unit the total is reached.
If the first addend is missing, students do not have a way to start an activity of counting
upward using their previous way of operating. They thus need to either reverse the order
of the addends (in effect, creating a missing second addend situation), or count down
from the total until they reach the second addend. Both of these operations are more
advanced and less likely to be available to many students due to insufficient abstraction
of the total as a composite unit containing of both addends as sub-composite units.
Additional considerations for the additive tasks are discussed next, in an attempt to clear
away distractions that might cause the assessment to give invahd results.
It is understood that some tasks may be much too simple for many students in the
upper grade bands, but to be thorough, they are included. It is also understood that the
34
concept of multiplicative reasoning assessed later in the instrument may not be present in
students in the earlier grades. Yet, the task is designed in such a way that if younger
students are able to reason multiplicatively, the numbers and situations should not be a
barrier.
The levels in each task were created to target values and situations that were
developmentally appropriate for students who are achieving at the grade-levels indicated.
The example seen in Figure 3.1, the Missing First Addend task, is differentiated to four
levels to accommodate grades K through 8. I chose the bouncy ball situation in the same
way that all situations were chosen, with an attempt to provide familiarity and context to
the problems. The numbers for each level (grade-band) were selected (a) in consultation
with Dr. Tzur, (b) using expert feedback from 15 teachers, and (c) guided by the work of
Sherin and Fuson (2005). The latter, empirical study focused on ways in which chosen
numbers and problem types affect an assessment. An effort was made not to repeat and
addition or multiplication facts, so that regardless of which grade level tasks were given,
or in which order, there would not be any repeats. Also I did not include consecutive
facts, such as 3 X 5 and 4 x 5, so that there would not be reliance on the previous
problems to answer the later problems.
For example, in the Missing First Addend Task (Figure 3.2), the lower grade
levels are given the addend 3 and a sum of 8, so that the total does not pass ten and
involve place value. These are numbers that younger students are likely to be familiar
with, yet avoid the facts that they might know, such as 5 + 5 = 10. For the oldest grade
band (7-8), the AMR uses the numbers 25 and 38 to discourage use of known facts and
encourage reasoning. It is unlikely that the student has the math fact 25 + 13 = 38in their
35
memory, as they may have 4 + 5 = 9, and therefore will make it more likely that they will
use additive reasoning to find the solution (e.g., 25+10-35, and 3 more is 38 — a strategy
indicating the student's operation on 13 by decomposing it strategically into 10 and 3).
Each task was designed with these considerations in mind, in an attempt to hone in on the
reasoning available without interference from memorized facts or inaccessible numbers
and situations.
Missing first addend task
Level 1, K-2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 3-4
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 5-6
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twenty-four (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 7-8
13. Anna has some bouncy balls. Her mom gives her twenty-five (25) more
bouncy balls. She then has thirty-eight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
Figure 3.1.Missing First Addend Task
There are some prevailing strategies commonly taught in schools to handle these
types of addition problems. The strategies seen in early trials of the problems were using
known doubles, such as 4 and 4 make 8, and building on them, so that 4+5 could be seen
as the double of 4 plus one. Another strategy is to 'break' or decompose the numbers to
make ten, so that 8+5 could be seen as 8+(2+3) = (8+2) + 3 = 10 + 3 = 13. Clearly, many
of these strategies that do not involve counting on allow students to use them in any
situation, and could not be avoided completely. A resourceful student can use these
36
strategies on any problem. In fact, this adaptability shows a high level of proficiency with
additive situations, particularly the operation on and with composite units, and thus
indicates that the student is ready to proceed.
These problems require students to be able to either use a modification of the
counting-on procedure, where they know to stop at the given total, and must find how
many more they need to count from the given addend. Alternatively, they can use
counting down, or beginning at the total and counting backwards until they reach the
known addend. The Missing Addend problems were created with the missing addend as
a quantity that is less than ten. As mentioned in the section on Counting On, this was
done because students are being asked to show their reasoning using figural or
manipulative means, so it was necessary that the missing portion be kept to a number that
would be reasonable for students to represent in this way. Only two of these tasks were
included, both considered Joining Problems by Carpenter et al. (1981). The reason is
twofold — ^the AMR was not intended to be a comprehensive assessment of additive
reasoning, and the study by Carpenter indicated that this deviation in problem structure
did not affect the solution methods of students. It is, intended that both tasks be
administered as a reliability measure. It is possible that a student could miss the first
problems, the easier missing second addend problem, for various reasons, but correctly
solve the more difficult second problem with a missing first addend.
Multiplicative Tasks
Multiplicative Double Counting, as described in the section on Multiphcative
Schemes in Chapter II, represents the first stage in the transition fi-om additive to
37
multiplicative thinking. This is the first and only multiplicative stage assessed by the
AMR in this prototype.
Due to the fact that Multiplicative Reasoning was the primary goal of the
development of this assessment, there are more tasks to lend reliability to this section.
The three tasks were all created with the unit rate, how many items per group, being less
than the number of composite units given in the problem. For example, in the first MDC
task for level 1, there are three cars per box and four boxes. The tasks were constructed
in this way to make it more conducive to the creation of a figural, pictorial or
manipulative solution. At the time of the defense of this thesis, it was suggested by Dr.
Heather Johnson, that the multiplicative tasks be adjusted to be consistent as to the
relative size of the unit compared to the unit rate. This additional expert feedback for
task adjustment was made and is reflected in the final instrument.
Task Refinement
Initial refinement of these tasks was done through feedback fi-om 15 teachers of
K-12 students. One suggestion they gave was to include picture cards for younger
students. A picture was included on the Composite Unit Iteration task, a frequent
suggestion, to help clarify the meaning of a pair of socks mentioned in the problem. I
decided not to include picture cards for the remainder of the problems, because it may
provide too much prompting via visible units. This would fail to show whether a student
has the reasoning or simply relies on the picture, counting Is while visually and silently
scanning the picture. I believed that part of the scheme necessary for students would be
the ability to create the pictorial or figural representation, and that providing a picture
might detract from their ability to show this crucial step in operating on units. It would
38
be appropriate for a kindergari:ener to be unable to solve even the additive reasoning
problems, so I did not feel that providing this scaffolding was necessary.
The remainder of the teachers' suggestions involved the numbers chosen for the
tasks. There were some repeats in the number families chosen in the initial draft, such as
4+5=9 being used three times on different problems. This was corrected in the later
version to include more addition facts, carefully avoiding the doubles and other patterns
described earlier. It was also suggested that a problem on the initial version involving
earrings be changed to a more gender-neutral item such as mittens. This change was
made to have the problem ask about socks, after much consideration as to an item that
students might have multiple pairs of at home.
After receiving the expert feedback from Dr. Tzur and the K-12 teachers, the
instrument was tested with fourth graders. As a result of that initial testing, the grade
bands were narrowed to accommodate students in a more differentiated way. The first
draft of this assessment included three levels, K-2, 3-5 and 6-8. During the initial testing
of the AMR it was found that some fourth grade students struggled with the values given,
and that the grade bands may have been too large. This prompted the fijrther
differentiation of levels in elementary grades to include only two grade levels instead of
one. The current structure includes levels for K-1, 2-3, 4-5 and 6-8.
Construction of Prompts
In this section, I will explain the way that prompts were developed for each of the
aforementioned tasks. These prompts are meant to assist a student who may not be at the
anticipatory stage in solving the problem with support, and thus gaining data about the
participatory or pre-participatory stage in which the student operates for the scheme being
39
tested. This part of the assessment, in which prompts are provided gradually, is designed
to give the assessor an idea of where the student is on the participatory continuum, and
what specific support they may need. This is important because a student who has no
prior knowledge of the reasoning at this stage will need to be approached differently than
one who is nearly anticipatory, and a nearly anticipatory student will likewise need
different learning experiences than one who is just beginning to be participatory. This
type of stage distinction was the goal of scripting the prompts to be given while the tasks
are being completed, instead of simply presenting the problem and prompts to the student
as typically occurs in a pencil and paper test.
As explained in the Fine-Grained Assessment section of Chapter II, the first step
is to administer the problem without prelude or prompts. This is intended to allow
students who have achieved an anticipatory stage for the scheme to demonstrate it. Using
the example of the Missing First Addend task above, the student would be given the
problem as is and asked to solve while explaining their thinking. If independently doing
so, it allows the assessor to infer construction of the scheme at the anticipatory stage.
Whether or not the problem is solved successfully, the student may then be
prompted to use figural, pictorial or manipulative means to prove their solution in an
attempt to understand the reasoning used by the student. This may be omitted sufficient
evidence was given that the student understands the concept, but it provides an
opportunity to delve deeper into what the student is thinking about the task. The use of
these means can also be a way for the student to self-prompt, or catch their own mistakes.
For a student who has used known facts, drawing or using manipulatives allow the
40
assessor to see if these facts are supported by reasoning, or simply reflect rote
memorization.
If the problem cannot be successfully solved independently, the assessor will then
begin to prompt the student in an attempt to determine whether the student has
constructed a participatory stage for the scheme as indicated by solving the problem with
support. The prompts are scripted purposely to provide the student information that
should help to orient their thinking to the activity involved in the scheme and thus to a
correct solution if they have begun forming that underlying conception. To the extent
possible, assessors are expected to avoid deviating significantly from these prompts, in
part because a teacher or researcher who works regularly with students may
unintentionally give additional information that would lead the student to score more
highly than is accurate.
An example of the prompting created for the Missing First Addend Task (Figure
3.2), which begins after the student has attempted to solve independently, is seen in
Figure 3.3. Some prompts were included to elicit specific reasoning from the student, if
the correct answer is given but sufficient reasoning is not.
41
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again.
o If at any time the student answers correctly using counting on or counting back, go
to the next problem.
If response is correct, b ut the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1 . How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
• If the student solves correctly after prompting, move to the next problem
Figure 3.3. Prompts for Correct Responses
A second level of prompts was developed for students who do not answer
correctly, seen in Figure 3.4, to determine if they have a high or low participatory level of
understanding, or perhaps no understanding of this stage at all.
If response is incorrect:
Ask student to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
• If they does not solve correctly with manipulatives, prompt students using the
following questions:
1 .How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
• If the student solves correctly after prompting, move to the next problem
• If the student cannot solve this problem after prompting, end testing.
Figure 3.4. Prompts for Incorrect Responses
The prompts here were designed to carefully lead a student through the
activity-effect thinking process that can help them arrive at a solution. The intent
42
of this stepwise approach is, again, to discern the specific level of the student
within the participatory stage. In the next section the choice of Administration
Method is discussed, with a rationale for the interview format.
Selection of Administration Method
To distinguish between the anticipatory and participatory stages in fine-grained
assessment, conducting a face-to-face interview is needed to provide the assessor with the
most specific feedback about what the student knows (Tzur, 2007). The format of the
assessment is a clinical interview, a technique often used to delve in to the thinking of
subjects, using what Zazkis and Hazzan (1998) called performance questions, followed
by "Why" questions. The interviews are to be conducted with one to three students at a
time, and are intended to give the students a task, observe their responses, and if needed,
prompt with questions that will clarify whether they are showing evidence of the scheme
or not (Ginsberg, 1981).
A clinical interview situation (Ginsberg, 1981) is best suited to this type of
assessment, because it allows the assessor to control and adapt the prompts and
information given to the student. The interview format provides the assessor with the
unique opportunity to ask students questions as they are solving, listen to what is being
verbalized, analyze and clarify figural representations created by the student in real time,
and discern any movements or quiet vocalizations that the student may produce.
While a paper and pencil task is efficient fi-om a time standpoint, it does not
provide the rich data that can be gleaned Irom an interview setting. Furthermore, in a
pencil and paper test, all of the information is present for the student to use, and it is
extremely difficult to discern whether the student was able to solve a problem
43
independently, whether they used information given in previous or subsequent problems,
what kind of reasoning was used, or whether they could have solved with a little more
support. Asking students to explain their answers in writing during a paper and pencil
test is another option, but it often provides inaccurate results (e.g., when the student's
first language is not English). Students can often explain verbally what they would not
write on a test, and may know more than they show on paper.
The tasks were ordered so that students would be unlikely to be able to complete a
task if they were unable to complete the previous task, a Guttman-like scale (Trochim &
Donnelly, 2008). This scaling is discussed further in Chapter IV. For that reason, it is
unnecessary for a student to complete the entire assessment. Rather, the assessor can
simply discontinue testing when the children being assessed reach a level they are
evidently unable to complete, even with prompts. As explained previously, the tasks
should begin at the Composite Unit Iteration task (pairs of socks), and then either move
forward to the multiplicative tasks if students reason independently, or begin the additive
reasoning tasks if they do not. One reason for using the Composite Iteration Task as a
starting point is that composite unit is a conception that is necessary throughout the tasks.
The ability to consider a group of Is as its own unit is something that a student would
need to be able to do to reason multiplicatively or additively, but the iteration of these
composite units is a more advanced concept than simply recognizing composite units.
This allows a 'middle ground' for students to begin at a level that is most appropriate for
them, without subjecting more advanced students to problems that are too easy for them,
and possibly losing their interest, or intimidating students who are unable to reason
multiphcatively. This can also save time in the administration procedure for both
44
teachers and students. Dr. Johnson suggested, with Dr. Alan Davis' support, this change
in task order at the time of thesis defense, and it was implemented in the Administration
Guide.
Students from time to time may struggle with the tasks not because of the
concept, but because of the numbers involved. To accommodate these students and
assess their reasoning, it is possible in the course of administering the test to allow the
student to respond to a task that is at the grade level below the one being assessed. For
instance, if a fifth grade student is struggling to complete the Multiplicative Double
Counting task, they could be administered the level 2 task, intended for second and third
graders. If the student could then answer the problem, it could be surmised that they
began forming the necessary multiplicative scheme, but may need some work to
generalize it for larger quantities. This flexibility was included so that the assessor can be
responsive to the needs of their students.
If or when a student is unable to complete the task with prompting and/or use of
manipulatives, the assessor terminates the testing. Such inability indicates that the
reasoning of the student has been exhausted and that they need not be tested any fijrther.
It is possible that a student who multiplies easily, and has some explanations as to what is
occurring may not score as highly as expected. This result might indicate that while the
student has a strong procedural knowledge or fact memorization, some of the prerequisite
schemes may not be solid enough to support their future learning. As discussed in
Chapter II, students will use available schemes to construct knowledge (Piaget, 1985),
but if not carefully monitored, may use schemes such as additive reasoning or part-of-a-
whole, which might not support them as well as multiplicative reasoning could. Next I
45
will explain how the Administration Guide was created to walk assessors through the
instrument as it is being given.
Development of Administration Guide
The administration guide is shown in Chapter IV, but this section will serve as a
brief description of the thinking behind its development. The guide was created to be
user- friendly, complete with scripted prompts and a description of what a person using
the assessment should look for in the child's solution to each problem. The task is listed
with each grade band and all prompts, in an attempt to provide for the assessor a clear
picture of what will happen during the assessment, and the options they have during
administration. It was also created with an eye toward some technological component
being involved in the future, and considering some decisions the technology would need
to make based on the student's response. The teacher or assessor can use the instrument
as is to pinpoint a student's understandings and create a fine-grained assessment (Tzur,
2007). Such assessment should be helpful to determine the next steps that need to occur
for the student, and help the assessor to complete the scoring guide described in the next
section.
Scoring Considerations
As with the administration guide, the Scoring Guide is seen in Chapter IV, but in
this section I will describe the thought process behind its development. The scoring of
this assessment needed to be both detailed and simple to complete so that teachers and
researchers can obtain the needed information while minimizing the time taken for the
assessment. Initially, I had envisioned the assessor scoring in real time, as the
administration proceeded. In testing of the instrument, however, it was noted by Mr.
46
Evan McClintock during this first administration that it might be simpler to have an
observer score as the assessor interacts with the children. This was a very useful
suggestion, and it was added as a possibility for the scoring process. It was also noted
during the testing process that if students are videotaped, as they were in our first trial,
scoring could happen later. I do not believe that this later scoring would be ideal,
however, as the scoring process may drive further questioning of the student in real time
that may not be possible after the fact. That being said, it may be used if necessary, and
if care is taken to thoroughly prompt the student.
The next chapter shows the final result of the work and refinement processes
described in this Methods Chapter. This instrument is still in development; but it is
presented in its complete and final form at the time of submitting this thesis to the
Graduate School of the University of Colorado Denver.
47
CHAPTER IV
RESULTS
This chapter presents the final version of the AMR, including tasks,
administration guide, student problem strips, and the scoring guide. Also included is a
discussion of the initial attempts made at discerning the reliability and validity of this
instrument.
Tasks
The tasks, seen in Figure 4. 1, are listed in the order of increasing conceptual
difficulty, but not necessarily in the order that they would be administered to a student in
an ordinary testing situation, because this order will vary depending on the result of each
problem. These tasks are also included in the Administration Guide, and in the Student
Problem Strips.
48
Composite Unit Iteration Task
1 . Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
Counting on task
Level 1, K-2
2. Lisa had five (5) pencils. She finds four (4) more pencils. How many pencils
does Lisa have?
Level 2, grades 3-4
3. Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
Level 3, grades 5-6
4. Lisa had twenty-four (24) pencils. She finds seven (7) more pencils. How many
pencils does Lisa have?
Level 3, grades 6-8
5. Lisa had fifty-seven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
Figure 4,1. Tasks
49
Missing second addend task
Level 1 K-2
6. Joseph has six (6) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has nine (9) pieces of gum. How many pieces of gum did
his friend give him?
Level 2, grades 3-4
7. Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
fiiend give him?
8. Level 3, grades 5-6
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces
of gum. Now Joseph has twenty-three (23) pieces of gum. How many pieces
of gum did his fi-iend give him?
9. Level 4, grades 6-8
Joseph has twenty-five (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirty-six (36) pieces of gum. How many
pieces of gum did his friend give him?
Missing first addend task
Level 1, K-2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 3-4
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 5-6
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twenty-four (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 7-8
13. Anna has some bouncy balls. Her mom gives her twenty-five (25) more
bouncy balls. She then has thirty-eight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
Figure 4.1. Tasks cont'd
50
III 1 aSK 1
T airai 1 1^ '>
j^evei 1, Jv-z
1 A
Tony has his toy cars in small boxes in his room. He has four (4)
boxes, and each box has three (3) cars in it. How many cars does Tony
nave m an
i_ievei z, graaes j-4
13.
Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have
in q119
m an
i_ievei J, graaes 3-o
lo.
Tony has his toy cars in small boxes in his room. He has six (6) boxes.
and each box has five (5) cars in it. How many cars does Tony have in
all?
Level 4, grades 7-8
17.
Tony has his toy cars in small boxes in his room. He has seven (7)
boxes, and each box has six (6) cars in it. How many cars does Tony
have in all?
muu 1 asK z
Level 1, Jv-z
JH/iid ndo live ) DasKcis ior ner ineiiuo. isne waiiis lo pui uirec )
pieces of candy in each basket. How many pieces of candy will Ella
need?
J^evei z, j-4
1
j_/Ua nda iivc ^^j ) DdaKcLa ioi nci iiicnua. one wdnLa LO pui ioui y-tj
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level J, 5-0
j_/Ud nda seven yi ) odaKeia ioi nei iiienua. one wdnLa lo puL ioui \^)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 4, 7-8
21.
Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Figure 4.1. Tasks cont'd
51
mDC Task 3
Level 1, K-2
22. Aaron wants to make three (3) birdhouses, and he needs two (2) nails
for each birdhouse. How many nails will he need in all?
Level 2, 3-4
23. Aaron wants to make six (6) birdhouses, and he needs four (4) nails for
each birdhouse. How many nails will he need in all?
Level 3, 5-6
24. Aaron wants to make eight (8) birdhouses, and he needs four (4) nails
for each birdhouse. How many nails will he need in all?
Level 4, 7-8
25. Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails
for each birdhouse. How many nails will he need in all?
Figure 4.1. Tasks cont'd
52
Assessment Protocol
Students are tested using the following protocol. The student(s) should be
provided with pencil and paper, or some electronic alternative that allows them to write
or draw only. A card containing the problem is handed to the student. These are
formatted and printed with the problem number for ease of reference. Instructions for the
cutting out and assembly of these problems strips are in Figure 4.2.
Student Problem Strips
• To use these strips, print out on cardstock and cut along the lines that border the tasks.
• Stack each group of problems (found on the same page) with the highest grade -band in
the back.
• This should result in a stack of strips that exposes the problem at the top of the strip.
• These can be left loose or stapled on the right or left into small booklets.
• The Composite Unit Iteration Task on this page is not part of any of the booklets.
1 . Composite Unit Iteration Task
Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
Figure 4.2. Student Problem Strips
53
2. Counting on task Level 1, grades K-2
Lisa had five (5) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
3. Counting on task Level 2, grades 3-4
Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
4. Counting on task Level 3, grades 5-6
Lisa had twenty- four (24) pencils. She finds seven (7) more pencils. How
many pencils does Lisa have?
5. Counting on task Level 4, grades 6-8
Lisa had fifty-seven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
Figure 4.2. Student Problem Strips cont'd
54
6. Missing Second Addend Task Level 1, grades K-2
Joseph has six (6) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has nine (9) pieces of gum. How many pieces of gum did his friend
give him?
7. Missing Second Addend Task Level 2, grades 3-4
Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
friend give him?
8. Missing Second Addend Task Level 3, grades 5-6
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has twenty-three (23) pieces of gum. How many pieces of
gum did his friend give him?
9. Missing Second Addend Task Level 4, grades 6-8
Joseph has twenty-five (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirty-six (36) pieces of gum. How many pieces
of gum did his friend give him?
Figure 4.2. Student Problem Strips cont'd
55
10. Missing First Addend Task Level 1, grades K-2
Anna has some bouncy balls. Her mom gives her three (3) more bouncy balls. She
then has eight (8) bouncy balls. How many bouncy balls did Anna have in the
beginning?
11. Missing First Addend Task Level 2, grades 3-4
Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna
have in the beginning?
12. Missing First Addend Task Level 3, grades 5-6
Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twenty-four (24) bouncy balls. How many bouncy balls did
Anna have in the beginning?
13. Missing First Addend Task Level 4, grades 6-8
Anna has some bouncy balls. Her mom gives her twenty-five (25) more bouncy
balls. She then has thirty-eight (38) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Figure 4.2. Student Problem Strips cont'd
56
14. Multiplicative Double Counting Task 1 Level 1, grades K-2
Tony has his toy cars in small boxes in his room. He has four (4) boxes,
and each box has three (3) cars in it. How many cars does Tony have in
all?
15.
Multiplicative Double Counting Task 1 Level 2, grades 3-4
Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have in
all?
16.
Multiplicative Double Counting Task 1 Level 3, grades 5-6
Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has five (5) cars in it. How many cars does Tony have in
all?
17.
Multiplicative Double Counting Task 1 Level 4, grades 6-8
Tony has his toy cars in small boxes in his room. He has seven (7) boxes,
and each box has six (6) cars in it. How many cars does Tony have in all?
Figure 4.2. Student Problem Strips cont'd
57
18. Multiplicative Double Counting Task 2 Level 1, grades K-2
Ella has five (5) baskets for her fiiends. She wants to put three (3) pieces
of candy in each basket. How many pieces of candy will Ella need?
19.
Multiplicative Double Counting Task 2 Level 2, grades 3-4
Ella has five (5) baskets for her friends. She wants to put four (4) pieces of candy
in each basket. How many pieces of candy will Ella need?
20.
Multiplicative Double Counting Task 2 Level 3, grades 5-6
Ella has seven (7) baskets for her friends. She wants to put four (4) pieces of
candy in each basket. How many pieces of candy will Ella need?
21.
Multiplicative Double Counting Task 2 Level 4, grades 6-8
Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Figure 4.2. Student Problem Strips cont'd
58
22. Multiplicative Double Counting Task 3 Level 1, grades K-2
Aaron wants to make three (3) birdhouses, and he needs two (2) nails for
each birdhouse. How many nails will he need in all?
23.
Multiplicative Double Counting Task 3 Level 2, grades 3-4
Aaron wants to make six (6) birdhouses, and he needs four (4) nails for each
birdhouse. How many nails will he need in all?
24.
Multiplicative Double Counting Task 3 Level 3, grades 5-6
Aaron wants to make eight (8) birdhouses, and he needs four (4) nails for each
birdhouse. How many nails will he need in all?
25.
Multiplicative Double Counting Task 3 Level 4, grades 6-8
Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails for
each birdhouse. How many nails will he need in all?
Figure 4.2. Problem Strips for Students cont'd
59
Once the card is given to the student, the assessor reads the problem aloud.
Clarification about what the problem is asking is permissible, as long as it does not point
students toward a solution. The student is asked to solve the problem by any means they
wish, thinking aloud as they do so. If students do not share their thinking during the
solution process, the assessor should ask the student after an answer is given what they
were thinking as they solved, encouraging making marks or drawings on their paper, or
using manipulatives to show their thinking.
Students should be encouraged to use writing, drawings or marks first before
using manipulatives. This is because using manipulatives, such as Unifix cubes, may
encourage students to count all instead of using a composite unit (Carpenter et al., 1981).
When students are representing the composite units, it is preferable if they can do so
without drawing in or building each of the Is needed to construct that unit. Treating the
unit as a whole, in and of itself, is important in the understanding of composite unit
(Tzur, et al., 2013). For example, if the child began drawing 6 circles with 3 dots in each,
the assessor should encourage him or her to first try solving the problem by just drawing
the first 1-2 circles and then just draw the circles without dots. Such an intervention is
done because students may count the Is even if their reasoning supports activity on the
composite units without the Is, simply because the Is are available to count. These
instructions are seen in the Administration Guide presented later in this chapter. The
manipulatives used could vary, but most commonly used are Unifix cubes and paper and
pencil for drawing pictures by the student. Students can solve problems with
manipulatives that they would not be able to solve without them, and so this intervention
is seen as similar to a prompt (Steffe, 1970).
60
The assessor should do her or his best to avoid indicating to the student whether
their answer or their thinking is correct or incorrect. The "burden of proof ' is on the
student, to be able to support their answer with reasoning, including figural or
manipulative means. This approach is needed to uphold the fme-grained assessment
technique (Tzur, 2007), which proceeds from asking the student to solve with the
minimum amount of support necessary. In order to determine whether the student is at an
anticipatory or participatory stage, it is crucial to allow the student to solve as
independently as possible, anticipating as much as they are able before prompts,
including those about correctness of solution, are given.
By following the guide, the assessor progresses through the questions using the
prompts, either moving down a level or using questioning to determine whether a student
may be able to solve a similar problem with different numbers. This will help teachers
determine whether what is lacking is the ability to reason about these tasks, or the
familiarity with the quantities presented. At the end of each task the assessor can find
directions about the next step in the process. Generally, if the problem was solved
correctly, the student moves to the next problem, if the student could not solve the
problem with prompting, testing is terminated. This is due to the attempt at a Guttman-
like organization (Trochim & Donnelly, 2008). This instrument is certainly not qualified
as using the Guttman scale, which would be based on data from previous administrations,
and not require that the student complete each item to give a reliable result. Details about
this process are seen in the Administration Guide in the next section.
61
Administration Guide
This section provides the complete version to date of the Administration Guide,
intended for use by teachers or other assessors, and showing the instructions, tasks,
prompts and things to look for at each stage. This document would guide the
administration of the assessment for students at any level. Using the guide, seen in
Figure 4.3, should direct teachers and other assessors in how to administer the
assessment, and help in the completion of the Scoring Guide seen in the following section
62
Administration Guide
Assessment for Multiplicative Reasoning
General Directions
Please read each task exactly as written. Students may also view the task and read it
on their own from the problem strip as you read it aloud.
Administration begins with the Composite Unit Iteration task, and from there will
move to either Counting On or Multiplicative Double Counting, depending on student
response. The progression is shown in the flow chart for Administration Protocol, and
is indicated in this guide.
The problem numbers are listed on the far left for easy reference.
Directions and numbers in task may be repeated indefinitely, and students may write
down important information or numerals as needed. Clarification about what is meant
by the problem is permitted, but students should do all solving. If possible, stick to
the given prompts.
Students may use scratch paper, and all marks made by students should be kept and
labeled with the problem number.
When administering these tasks, ask the student to explain their thinking, and record
everything the student does on the scoring sheet.
You may use questions such as:
• "What were you thinking as you solved this problem?"
• "What strategies did you use?"
• "Did you picture something in your head to help you solve this problem?"
Begin with the level appropriate for the child's grade level. If they are not successful
at this task, move down one level and test again.
Please explain to the student that you will be recording their responses and their work
so that we can learn more about how they solve problems.
Do not indicate to the student whether they are correct or incorrect as they work.
Figure 4.3. Administration Guide
63
Administration Sequence
Composite Unit
Iteration Taslc
(Problem 1)
Incorrect
Solution
Z
Additive Tasks
(Problems 2-13)
Counting On
(Problems 2-5)
Missing Addend Tasks
(Problems 6-13)
Correct
Solution
Multiplicative Tasks
(Problems 14-25)
1
Multiplicative Double
Counting
(Problems 14-25)
I
If students solve the addend tasks
correctly, begin the
Multiplicative Tasks. It may be
helpful to begin at a lower grade
band if the students cannot reason
with the numbers provided.
Figure 4.3, Administration Guide cont'd
64
Composite Unit Iteration Task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
(Clarify the meaning of "pair".)
1 . Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
• If the student counts one by one, and not by composite units of two, ask if they
can use the way the socks are grouped into pairs.
• If the students cannot solve after this prompt, begin with the Counting On Task
(problem 2).
• If the student is able to iterate the socks to find that there are 16 in all, skip to the
Multiplicative Double Counting Tasks (problem 14).
Counting on task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level 1, K-2
2, Lisa had five (5) pencils. She finds four (4) more pencils. How many pencils
does Lisa have?
Level 2, grades 3-4
3, Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
Level 3, grades 5-6
4, Lisa had twenty-four (24) pencils. She finds seven (7) more pencils. How many
pencils does Lisa have?
Level 3, grades 6-8
5, Lisa had fifty-seven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
o If the student has answered correctly, and shown evidence of counting on, go to the
next problem.
Figure 4.3. Administration Guide cont'd.
65
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
tirst composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the progress of the student's progress, move down to a lower
grade band and test again.
o If at any time the student answers correctly using counting on, go to the next
problem.
If response is correct, b ut the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1 .How many did she have in the beginning?
2. H0W many did she find?
3. Can you use the number she had in the beginning as a starting place to see how
many in all? How?
• If the student uses counting on after prompting, move to the next problem.
If response is incorrect;
Ask students to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
• If solved correctly with manipulatives, go to the next problem.
• If you believe that the number values are impeding the progress of the student,
move down to a lower grade band and test again.
• If they do not solve correctly, prompt students using the following questions:
• How many did she have in the beginning?
• How many did she find?
• Can you use the number she had in the beginning as a starting place to see how
many in all? How?
• If the student uses counting on after prompting, move to the next problem
• If the student cannot solve this problem after prompting, end testing.
66
Additive Tasks
Missing Addend Tasks
• Administer both of these tasks, regardless of the results, to assess the
additive reasoning of the student.
Missing second addend task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level 1 K-2
6. Joseph has six (6) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has nine (9) pieces of gum. How many pieces of gum did
his friend give him?
Level 2, grades 3-4
7. Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
friend give him?
8. Level 3, grades 5-6
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces
of gum. Now Joseph has twenty-three (23) pieces of gum. How many pieces
of gum did his friend give him?
9. Level 4, grades 6-8
Joseph has twenty-five (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirty-six (36) pieces of gum. How many
pieces of gum did his friend give him?
• If the student uses counting on or counting back, move to the next problem.
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again.
o If at any time the student answers correctly using counting on, go to the next
problem.
67
If response is correct, b ut the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1 . How many did he have in the beginning? How many does he have in the end?
Can you use the number in that he had in the beginning as a starting place see
how many more he got from his friend? How?
2. Can you use the number he had in all and count backwards to get to find out
how many he got from his friend? How?
If the student solves correctly after prompting, move to the next problem.
If the student cannot solve this problem after prompting, end testing.
If response is incorrect:
Ask student to explain their thinl^ing on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
1 . If solved correctly with manipulatives, go to the next problem.
2. If you believe that the number values are impeding the student, move down
to a lower grade band and test again.
3. If they do not solve correctly, prompt students using the following questions:
4. How many did he have in the beginning? How many does he have in the
end? Can you use the number in that he had in the beginning as a starting
place see how many more he got from his friend? How?
5. Can you use the number he had in all and count backwards to get to find out
how many he got from his friend? How?
• If the student solves correctly after prompting, move to the next problem
• If the student cannot solve this problem after prompting, end testing.
Missing first addend task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level 1, K-2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 3-4
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 5-6
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twenty-four (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 7-8
13. Anna has some bouncy balls. Her mom gives her twenty-five (25) more
bouncy balls. She then has thirty-eight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
68
• If the student uses counting on or counting back, end assessment.
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again.
o If at any time the student answers correctly using counting on or counting back, go
to the next problem.
If response is correct, b ut the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1 . How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
• If the student solves correctly after prompting, move to the next problem
If response is incorrect:
Ask student to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
• If they does not solve correctly with manipulatives, prompt students using the
following questions:
1 .How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
• If the student solves correctly after prompting, move to the next problem
• If the student cannot solve this problem after prompting, end testing.
69
Multiplicative Tasks
Multiplicative Double Counting Tasks
There are three tasks in this section. A successful solution to two out of three is
considered evidence that the student has access to the Multiplicative Double Counting
Scheme, and testing should be concluded when the student has successfully completed
two.
mDC Task 1
Level 1, K-2
14. Tony has his toy cars in small boxes in his room. He has four (4)
boxes, and each box has three (3) cars in it. How many cars does Tony
have in all?
Level 2, grades 3-4
15. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have
in all?
Level 3, grades 5-6
16. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has five (5) cars in it. How many cars does Tony have in
all?
Level 4, grades 7-8
17. Tony has his toy cars in small boxes in his room. He has seven (7)
boxes, and each box has six (6) cars in it. How many cars does Tony
have in all?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• If the student gives an answer without units, ask what the number tells. Is it number
of cars? Number of boxes?
• If correct, and using mDC strategy to find the answer, proceed to mDC task 2.
• If they use counting all, or known math facts, prompt student using these
questions:
1 . How many cars in one box?
2. Repeat original question
3. How many cars would be in two boxes?
4. Repeat original question
Figure 4.3. Administration Guide cont'd
70
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• As soon as the student is able to give the answer correctly, move to mDC task 2.
• Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1 . How many cars in one box?
2. Repeat original question
3. How many cars would be in two boxes?
4. Repeat original question
Regardless of whether the student answers correctly or incorrectly, go on to mDC
Task 2.
mDC Task 2
Level 1, K-2
18. Ella has five (5) baskets for her friends. She wants to put three (3)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 2, 3-4
19. Ella has five (5) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 3, 5-6
20. Ella has seven (7) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 4, 7-8
21. Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• If the student gives an answer without units, ask what the number tells. Is it
number of baskets? Number of candy pieces?
• If correct, and using mDC strategy to find the answer, conclude testing.
• If they use counting all, or known math facts, prompt student using these
questions:
1 .How many candy pieces in one bag?
2. Repeat original question
3. How many candy pieces would be in two bags?
4. Repeat original question
Figure 4.3. Administration Guide cont'd
71
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• As soon as the student is able to give the answer correctly, move to mDC task 3.
• Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1 . How many candy pieces in one bag?
2. Repeat original question
3. How many candy pieces would be in two bags?
4. Repeat original question
• If both mDC tasks were answered correctly, conclude testing
• If one mDC task is answered correctly, administer mDC Task 3.
• If both mDC tasks were answered incorrectly, conclude testing.
mDC Task 3
Level 1, K-2
22. Aaron wants to make three (3) birdhouses, and he needs two (2) nails
for each birdhouse. How many nails will he need in all?
Level 2, 3-4
23. Aaron wants to make six (6) birdhouses, and he needs four (4) nails for
each birdhouse. How many nails will he need in all?
Level 3, 5-6
24. Aaron wants to make eight (8) birdhouses, and he needs four (4) nails
for each birdhouse. How many nails will he need in all?
Level 4, 7-8
25. Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails
for each birdhouse. How many nails will he need in all?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• If the student gives an answer without units, ask what the number tells. Is it
number of nails? Number of birdhouses?
• If correct, and using mDC strategy to find the answer, conclude testing.
• If the student uses counting all or known math facts, prompt student using these
questions:
1 . How many nails does he need for one birdhouse?
2. Repeat original question
3. How many nails does he need for two birdhouses?
4. Repeat original question
Figure 4.3. Administration Guide cont'd
72
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
• As soon as the student is able to give the answer correctly, conclude testing.
• Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1 . How many nails does he need for one birdhouse?
2. Repeat original question
3. How many nails does he need for two birdhouses?
4. Repeat original question.
End of Test
Figure 4.3. Administration Guide cont'd
Scoring Guide
Scoring of students' reasoning based on their solutions, including both answer and
explanation, for the tasks can optimally occur during the interview process, either by the
assessor or another observer. The assessment is meant to be adapted to what is seen in
student responses (e.g., counting Is or composite units, visible or figural, etc.), so that
care is taken to administer appropriate tasks to appropriate students. While the time and
care needed to do this type of assessment is significant, the benefits of having real-time
scoring and the ability to probe and question a student as they are working gives a better
understanding of the thinking and level of reasoning being employed by the student
(Ginsberg, 1981; Steffe, 2002). As students complete tasks, the assessor will record the
response on the scoring guide seen in Figure 4.4, including but not limited to the use of
figural items, manipulatives, drawings, or computational procedures. The extra space
provided in the middle of each page is intended for assessor's note taking.
73
Scoring Guide
Assessment for Multiplicative Reasoning
Grade Level of Student
Composite Unit Iteration Task (CU)
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□Incorrect Response with
prompts.
Pre -Participatory for this
scheme
Figure 4.4. Scoring Guide
74
Counting-on task
Indicate any procedures that students may use, including but not
limited to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□Incorrect Response with
prompts.
Pre-Participatory for this
scheme
Figure 4.4. Scoring Guide, cont'd.
75
Missing Second Addend (MSA)
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□incorrect Response with
prompts.
Pre -Participatory for this
scheme
Figure 4.4 Scoring Guide cont'd
76
Missing First Addend
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□incorrect Response with
prompts.
Pre -Participatory for this
scheme
Figure 4.4. Scoring Guide, cont'd
77
Multiplicative Double Counting 1 (mDCl)
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□Incorrect Response with
prompts.
Pre-Participatory for this
scheme
Figure 4.4. Scoring Guide, cont'd.
78
Multiplicative Double Counting 2 (mDC2)
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□Incorrect Response with
prompts.
Pre -Participatory for this
scheme
Figure 4.4. Scoring Guide, cont'd.
79
Multiplicative Double Counting 3 (niDC3)
Indicate any procedures that students may use, including but not limited
to the following:
□ Counts on fingers
□ Draws or uses marks
□ Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
□ Other behaviors (please describe)
Student Response
Stage Indicated by this Result
□ Correct Response to grade
level problem without
prompts.
Anticipatory at grade level for
this scheme
□Correct Response to grade
level problem with prompts.
Participatory at grade level for
this scheme
□Correct Response to below
grade level problem without
prompts.
Anticipatory, but below grade
level for this scheme
□Correct Response to below grade
level problem with prompts.
Participatory and below grade
level for this scheme
□Incorrect Response with
prompts.
Pre-Participatory for this
scheme
Figure 4.4. Scoring Guide.cont'd.
80
Using the Scoring guide should allow assessors to determine at which stage,
participatory or anticipatory, students seem to operate in each scheme tested. It may also
help to guide prompts during the interview process, as it becomes clear the evidence
needed to prove competence at each level.
Reliability and Validity
The AMR is work-in-progress in its preliminary stages. It has not yet been fully
tested for reliability or validity; however, attempts made to increase these two measures
of the instrument are discussed here. The validity measures taken in the development of
the AMR were based upon content validity according to expert opinion, both classroom
teachers and researchers. The items were developed under the advisement of Dr. Ron
Tzur of the University of Colorado, Denver. His guidance focused on ensuring that the
tasks were created to reflect the theoretical stances described in Chapter 2. Using
information about the numbers that would be most appropriate at each grade band, and
tying the tasks closely to the six Schemes of Multiplicative Reasoning discussed earlier,
the items are believed to be good indicators of the reasoning at each level. As explained
in Chapter III, these items were refined both by feedback fi-om teachers and through
informal testing with fourth-grade students. I was able to observe students as they
participated in the assessment, and also received feedback from the reports of Dr. Tzur
and Mr. McClintock on later administrations of the tasks I was unable to witness.
Additionally, I solicited critique by way of a Feedback Form from classroom teachers, as
discussed in context in Chapter IV. The Form used to collect this data is found in the
Appendix to this thesis.
81
Another reliability measure is the two-fold problem delivery of verbal directions
and printed student problem strips shown earlier in this section. These seemingly
redundant steps are to remove as many barriers to the students' interpretation of the
problem as possible. Students who have difficulty interpreting verbal questions have
access to the printed problem, while those who have difficulty with reading have the
verbal instructions read by the assessor. The ability to refer to the card and clarify
numbers or any other aspect of the problem, as well as review any information, was
deliberately included to ensure that incorrect solutions were not due to misunderstandings
or difficulty remembering the information given in the tasks.
The choice of numbers for each problem type and each grade level was
purposeful, as supported by the work of Sherin and Fuson (2005). The strategy used by
students in that study varied depending on the numbers and context used. It showed that
the choice of each number used in a task must be made mindfully to elicit the strategy for
which it is designed. If the numbers and situations for these problems were not chosen
carefully, students could use some learned strategies, or known facts, that do not
necessarily reflect their understanding of composite unit, such as how to count on. Sherin
and Fuson also note that the only route to this clear, complex, flexible knowledge is the
passage through earlier stages, through which they develop strategies rooted in
understanding. Clearly, an understanding of the problem is an important facet, and was
treated as such using these methods, to provide a reliable assessment.
Because the focus of this thesis was on multiplicative reasoning, there are three
tasks to add validity to this portion of the assessment. If a student is able to correctly and
independently solve two tasks in a row, testing is terminated, and the student is
82
considered to have access to the scheme for Multiplicative Double Counting. If,
however, the student solves only one out of the first two tasks independently and
correctly, the third task is administered to determine whether the student can be
considered to have this scheme, or at which stage. A score of two out of three correct
would indicate the student's readiness to move to the next level of multiplicative
reasoning.
These steps were taken to further increase reliability within the given time and
resources available, and could certainly help to improve the AMR further in future
endeavors. The following section includes a discussion of the implications and benefits
of the development of the AMR, as well as a further discussion of the limitations
indicated here.
83
CHAPTER V
DISCUSSION
Contributions to Research and Practice
This section will focus on the ways that the work seen in Chapter IV can benefit
the field of mathematics education in terms of research and classroom teaching. The
ability to test for and determine not only the multiplicative schemes available for students
but also the anticipatory or participatory stages within each scheme could be a great
benefit to teachers and researchers alike. In both the classroom and the field of research,
there is a need to find out what students understand about multiplicative reasoning, and
whether they have constructed a scheme at the participatory or anticipatory stage.
A classroom teacher could use this instrument to determine what students need to
be able to move ahead, into multiplicative reasoning, or whether they are solid in their
initial stage of multiplicative reasoning, and are ready to move on to more advanced
topics. For example, a student could be presented with the Missing Second Addend task
for the 3-4 grade -band. It reads: "Joseph brings eight (8) pieces of gum to school. His
fiiend gives him some more pieces of gum. When Joseph counts them all, he finds that
he has fifteen (15) pieces of gum. How many pieces of gum did his friend give him?"
The student's response allows a teacher to distinguish whether a students is at the
anticipatory or participatory stage of using composite units within a given, encompassing
composite unit. If at the participatory, (i.e., can only solve the problem with prompting),
teaching would then focus on moving the student toward an anticipatory stage, using her
current knowledge and making prompting explicit so that she will begin to self-prompt.
However, if the student is at the anticipatory stage (can solve independent of any
84
prompting while using strong reasoning), teaching can move forward to missing second
addend tasks, by using the missing first addend concepts as a bridge to these missing
second addend concepts. It may also indicate moving ahead to teaching mDC while
keeping in mind to focus on the student's way of reasoning with Is vs. with composite
units.
In addition, the assessment should be flexible enough to include a wide range of
learners. An advanced first grader could be assessed for compaction of curriculum or
advancement to a more challenging math group based on the AMR results. A struggling
seventh grader might be able to receive needed intervention based on results that show he
or she is not able to perform Multiplicative Double Counting. A fifth grade student who
does have the availability of the appropriate scheme, but does not compute fluently with
whole numbers, could test at a lower grade band and show this reasoning in spite of their
computational difficulties.
I see this awareness as essential for teachers, because the connection between
multiplicative reasoning and topics such as fractions is not always a focus of instruction
in those topics, though it might be a way to improve it. Fractions are multiplicative
quantities because they require a reference to the unit. When fractions are an extension
of multiplication, the idea that there are multiple ratios to consider in multiplication
(Confrey and Smith, 1995) can help create a bridge from the whole number to the
fractional concepts. Some of the ratios indicated in 5 x 7 are each composite unit of five
to seven ones, as the relation of the seven ones in one group and the five groups to the
thirty-five ones in total. This understanding of multiplication as multiple ratios builds the
85
foundation for students to understand fractional quantities in a way that repeated addition
does not.
When the meaning of five groups of seven is conceptualized by the student, the
understanding that 2/3 means that there are some units, of which the whole is three times
as much, and that we are considering two of these units, as 2 * 1/3 or 2 'groups' of 1/3,
can equip a student to compare relative values of these fractions. This thinking also
allows a student to understand why in a problem such as 2/3 + 4/5, we must use a
common measure for operating on both thirds and fifths. When students recognize thirds
and fifths as different units, and see this as a way to build upon their knowledge of
multiplication to iterate those units, the search for a common measure ('denominator') is
no longer some mysterious invention of teachers of mathematics, but a necessary
operation of multiplicative unit coordination needed to find common units. Lamon
(1996) saw a similar conception being built in students who use equal sharing tasks to
develop their fraction concepts. When students began to see certain quantities as
equivalent, they were using multiplicative rather than additive concepts to understand the
size of the portion. Being able to see and assess this reasoning could go a long way for
classroom teachers seeking to improve the teaching and learning of fractions and/or
multiplicative reasoning in their classrooms.
In research the AMR may be used for a similar purpose, or to find correlations
between multiplicative reasoning and other types of mathematical conceptions. For
example, it could be investigated whether a student who has a strong knowledge of area
and perimeter was more equipped with multiplicative reasoning than one who does not.
The AMR could be used to quickly determine the level of a student and then use it to
86
correlate with other factors. There are some limitations to this instrument, however, and
they are discussed next.
Limitations
While the AMR provides benefits for teachers and researchers, it is important to
be aware of its limitations. It is understood that the time involved in completing an
assessment of this type is limiting for most classroom situations. One way to decrease
the amount of time needed for testing is to administer, as a written test, the portion that
the teacher beheves the student has already constructed as a solid conception. In this way
the AMR's tasks function as a preliminary screener. For example, if a third grade teacher
believes students are proficient in missing second addend problems, the counting on and
missing second addend section could be given to students simply as a written test. The
students who show clearly that they are anticipatory for these tasks can begin with the
missing first addend for the interview. Those who did not show that they are anticipatory
could then be tested further, by interview, to determine if they may be participatory for
that stage or do not yet have the conception at all. As explained in Chapter II, the
distinction between anticipatory and participatory stages requires the task to be first given
without prompts, to determine whether the students can anticipate the entire process, and
if not, with prompts to assess for a participatory level of understanding.
The AMR is a prototype, and only addresses the first scheme in the multiplicative
reasoning continuum delineated by Tzur et al. (2013), namely. Multiplicative Double
Counting. This was done for simplicity, and to allow testing and feedback to determine
the foci of later instruments. The remaining stages of multiplicative reasoning could be
87
created in similar ways, and would show where more advanced students fall on the
continuum of multiplicative reasoning.
Instead of progressing from mDC to more advanced multiplicative schemes, the
AMR instrument includes schemes that are prerequisite to multiplicative reasoning, such
as counting on. These were included to determine the extent to which students have
constructed composite units needed for their journey from additive to multiplicative
reasoning, or are still developing the understandings on which additive reasoning should
be built (Steffe, 1994). In a study conducted with 5^'' grade students designated by their
school system as learning disabled, who did not reason multiplicatively (Tzur, 2010), it
was found that one of the major impediments to their progress was the lack of a concept
of number as an abstract, symbolic, composite unit.
The AMR is meant to begin with these prerequisite skills to save teachers time
when doing assessments. Clearly the additive portion of the test is not comprehensive
enough to account for all aspects of additive reasoning and all of the different types of
problem structures and what they may reveal about student thinking. Rather, the additive
portion was designed to give a glimpse into the schemes available to students, and
whether they have the conceptual prerequisites needed to engage in and learn to reason
multiplicatively. With this in mind, the assessment only covers counting on, missing
second addend problems with the missing portion being smaller, and a missing first
addend with the missing potion being smaller. It is understood that this spans a limited
range in the additive abilities of students, but does so while providing essential
information about the students' use and coordination of units. An assessor who finds that
students are in this general stage of reasoning and would like to further pinpoint their
88
development, would need to consult a more comprehensive additive reasoning
assessment.
It is also important to note that this instrument has never been tested in its
entirety. Significant changes were made following the initial testing, up to and including
the defense phase. The following section suggests possibilities for future research, some
of which address the aforementioned limitations.
Implications for Future Research
A continuation of the assessment to include the remaining items in the framework
for multiplicative reasoning, outlined by Tzur et al. (2013), would be helpful to teachers
who need to be able to assess all schemes, and stages within them, up to the point in
which they may be ready to learn fractions. In particular, items could be created for the
remaining schemes in this framework: Same Unit Coordination (SUC), Unit
Differentiation and Selection (UDS), Mixed Unit Coordination (MUC), Quotitive
Division (QD), and Partitive Division (PD), so the assessment is expanded for the entire
scope of these schemes. When a student has progressed through all of these schemes,
they may be considered proficient in multiplicative reasoning.
Additionally, the prototype of the AMR was written such that a computer-based
assessment could be built to either complete the preliminary screening questions, used to
decide where a student might need to begin testing, or as a tool to help the assessor. A
software program that could record responses of students, coded by the assessor, and
indicate the next task and level that should be administered, could be a valuable tool. For
example, if the student answers a question correctly with strong reasoning, the
technological component could advance the student to the next appropriate question. It
89
seems unlikely that the assessment could be productively administered without the
human interview component, due to the importance of the interaction between the
assessor and student, and the careful attention to student thinking gleaned from bodily
gestures and whispers, which may not be detected, let alone understood, by existing
technology.
Additionally, teachers may want to know how to meet student needs once their
reasoning, both scheme and stage, has been assessed. This is an excellent opportunity for
future work, and would greatly assist teachers in using the data obtained to foster the
understandings that they now know students need to prepare them for future work.
Concluding Remarks
This project was bom out of my own finstration, as a middle school teacher, with
the low level of knowledge that my students often have about fractional quantities. Too
often I saw that though students gained procedural knowledge with repetition and
practice, their conceptual understanding remained weak, despite activities that I designed
to try to build these conceptions. I truly believe that for teachers and other educators to
first and foremost become aware of the gaps in reasoning that cause students in middle
school, high school and even higher education to not only dread, but show a limited
understanding of the meaning of fraction, is key. A teacher's possession of both this
awareness and the knowledge about schemes used for reasoning multiplicatively, and of
anticipatory and participatory stages within each scheme, could go a long way towards
improving the teaching and learning of fraction and ratio concepts. Further, I am glad as
an educator to have an instrument that allows me to begin assessing the underlying
multiphcative reasoning that my students may or may not have. I foresee more
90
productive lessons when students are equipped with the necessary conceptual
prerequisites with which to construct fraction laiowledge, as opposed to asking them to
operate on conceptions that they are not able to access due to incomplete multiplicative
schemes.
91
REFERENCES
Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. Acquisition of
mathematics concepts and processes, (pp. 91-126). New York: Academic Press.
Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. A. (1984). Order and equivalence of rational numbers:
A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323-341.
Bonato, M., Fabbri, S., Umilta, C, & Zorzi, M. (2007). The mental representation of numerical fractions:
Real or integer? Journal of Experimental Psychology-Human Perception and Performance,
33(6), 1410-1419.
Bright, G. W., Joyner, J. M., & Wallis, C. (2003). Assessing proportional thinking. Mathematics
Teaching in the Middle School, 9(3), 166-172.
Bryant, B. R., & Maddox, T. (1996). Using alternative assessment techniques to plan and evaluate
mathematics instruction." Ariel, 129, 6-1.
Carpenter, T. P., Hiebert, J., & Moser, J. M. (1981). Problem structure and first-grade children's initial
solution processes for simple addition and subtraction problems. Journal for Research in
Mathematics Education, 12 (1), 27-39.
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5.
Journal for Research in Mathematics Education, 27(1), 41-51.
Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of
exponential functions. Journal for Research in Mathematics Education, 26(1), 66-86.
Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a). "Rational Number Project: Fraction Lessons for the
Middle Grades: Level 1." Dubuque, lA: Kendall/Hunt Publishing.
Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). "Rational Number Project: Fraction Lessons for the
Middle Grades: Level 2." Dubuque, lA: Kendall/Hunt Publishing.
De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders' strategies for
solving addition and subtraction word problems. Journal for Research in Mathematics Education,
18(5), 363-381.
Dehaene, S. (1997). The number sense: How the mind creates mathematics. NY: Oxford University.
Dewey, J. (1938). Experience and education. New York: Collie
Erlwanger, S. H. (1973). Benny's Conception of Rules and Answers in IPl Mathematics. Journal of
Children's Mathematical Behavior, 1(2), 7-26.
Fan, L., & Zhou, L. (2006). How Have Chinese Students Performed in Mathematics? A Perspective
from Large-Scale International Mathematics Comparisons. In L. Fan, N.-Y. Wong, J. Cai and S.
Li (Eds.), How Chinese Learn Mathematics: Perspectives from Insiders (pp. 3-25).
92
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving
verbal problems in multiplication and division. Journal for research in mathematics education, 3-
17.
Fuson, K. C, & Willis, G. B. (1988). Subtracting by counting up: More evidence. Journal for
Research in Mathematics Education, 19(5), 402-420.
Ginsburg, H. (1981). The clinical interview in psychological research on mathematical thinking:
Aims, rationales, techniques. For the learning of mathematics, 1(3), 4-11.
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2008). Highlights from
TIMSS 2007. US Department of Education, Institute of Education Sciences, National Center for
Education Statistics, http://nces. ed. gov/pubs2009/2009001. pdf.
Hartnett, P., & Gelman, R. (1998). Early understandings of numbers: paths or barriers to the construction
of new understandings? Learning and instruction, 8{A), 341-374.
Jacob, L. & Nieder, A. (2009) Notation-Independent Representation of Fractions in the Human Parietal
Cortex. The Journal of Neuroscience, 29(14), 4652-4657
Jacob, L., Willis, S.G., 2003, The development of multiplicative thinking in young children. In
Mathematics Education Research: Innovation, Networking, Opportunity: Proceedings of the 26th
Annual Conference of the Mathematics Education Research Group of Australia, 06/07/2003 to
10/07/2003, Deakin University, Melbourne Vic Australia, pp. 460-467.
Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal
for Research in Mathematics Education, 27(2), 170-193.
Levine, S. C, Jordan, N. C, & Huttenlocher, J. (1992). Development of calculation abilities in young
children. Journal of Experimental Child Psychology, 53(1), 72-103.
Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal
knowledge. Journal for Research in Mathematics Education, 26 (5), 422-441 .
McClintock, E. (Personal Communication, May 13, 2012)
Medina, J. (2008). Brain rules: 12 principles for surviving and thriving at work, home, and school.
Seattle, WA: Pear.
Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children's intuitive models of multiplication and
division. Journal for Research in Mathematics Education, 28(3), 309-330.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics
Reston, VA: NCTM.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National
Mathematics Advisory Panel. US Department of Education.
Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and
implications of whole number bias. Educational Psychologist, 40(1), 27-52.
93
Norton, A., & Hackenberg, A. J. (2010). Continuing research on students' fraction schemes. In L.P. Steffe
and J. Olive (Eds.), Children 's Fractional Knowledge (pp. 341-352). New York, NT:Springer
US.
Olive, J. (2001). Children's number sequences: An explanation of Steffe's constructs and an extrapolation
to rational numbers of arithmetic. The Mathematics Educator, 11(1), 4-9.
Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual
development (T. Brown & K.J. Thampy, Trans.). Chicago: The University of Chicago.
Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize
it and how can we represent it? Educational Studies in Mathematics, 26(2-3), 165-190.
Raykov, T.,Marcoulides, G. A. (201 1). Introduction to Psychometric Theory. Taylor & Francis. Retrieved
1 1 June 2013, from <http://lib.myilibrary.com?ID=303716>
Sherin, B., & Fuson, K. (2005). Multiplication strategies and the appropriation of computational
resources. Journal for research in mathematics education, 36(4), 347-395.
Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2012). Fractions: the new frontier for theories of
numerical development. Trends in cognitive sciences, 17 (I), 13-19.
Siegler, R. S., Thompson, C. A., & Schneider, M. (201 1). An integrated theory of whole number and
fractions development. Cognitive psychology, 62(4), 273-296.
Siemon, D., & Breed, M. (2010, May). Assessing multiplicative thinking using rich tasks. In Annual
Conference of the Australian Association for Research in Education. (Vol. 13). Downloaded from
http://publications.aare.edu.au/06pap/sie06375.pdf
Siemon, D., Breed, M., & Virgona, J. (2005). From additive to muhiplicative thinking-the big challenge
of the middle years. In Proceedings of the 42nd Conference of the Mathematical Association of
Victoria.
Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment
framework for multiplicative thinking.
Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin (Sage).
Stafylidou, S., & Vosniadou, S. (2004). The development of students' understanding of the numerical
value of fractions. Learning and Instruction, 14(5), 503-518.
Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual
Differences, 4(3), 259-309.
Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of
Mathematical Behavior, 20, 1-41.
Steffe, L. P., & Cobb, P. (1998). Multiplicative and divisional schemes. Focus on Learning Problems in
Mathematics, 20(1), 45-62.
Steffe, L. P. (1970). Differential performance of first-grade children when solving arithmetic addition
problems. Journal for Research in Mathematics Education, 1 (3), 144-161.
94
Steffe, Leslie P. (2002). The constructivist teaching experiment: Illustrations and implications. In Radical
constructivism in mathematics education, (pp. 177-194). Netherlands: Springer.
Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental
Research. Dordrecht, The Netherlands: Kluwer.
Streefland, L. (1993). Fractions: A realistic approach. In Carpenter, Thomas P., Fennema, Elizabeth, and
Romberg, Thomas A. (Eds.), Rational numbers: An integration of research, 289-325. New York,
NY :Routledge.
Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G.
Martin & D. Schifter (Eds.), A research companion to Principles and standards for school
mathematics (pp. 95-1 13). Reston, VA:
Torbeyns, J., Ghesquiere, P., & Verschaffel, L. (2009). Efficiency and flexibility of indirect addition in
the domain of multi-digit subtraction. Learning and Instruction, 19{\), 1-12.
Trochim, W. M., & Donnelly, J. P. (2008). Research methods knowledge base. Mason, OH:
Atomic Dog/Cengage Learning.
Tzur, R. (2004). Teacher and students' joint production of a reversible fraction conception. Journal of
Mathematical Behavior, 23, 93-114.
Tzur, R. (2007). Fine grain assessment of students' mathematical understanding: Participatory and
anticipatory stages in learning a new mathematical conception. Educational Studies in
Mathematics, 66(3), 273-291.
Tzur, R., Johnson, H. L., McClintock, E., Kenney, R. H., Xin, Y. P., Si, L., .... Jin, X. (2013).
Distinguishing schemes and tasks in children's development of multiplicative reasoning. PNA,
7(3), 85-101.
Tzur, R., Johnson, H., McClintock, E., Xin, Y. P., Si, L., Kenney, R., et al. (2013). Children's
development of multiplicative reasoning: A schemes and tasks framework. In T.-Y. Tso (Ed.),
Proceedings of the 36th conference of the International Group for the Psychology of
Mathematics Education (Vol. 4, pp. 155-162). Taipei, Taiwan: National Taiwan Normal
University.
Tzur, R., & Lambert, M. A. (201 1). Intermediate participatory stages as Zone of Proximal Development
correlate in constructing counting-on: A plausible conceptual source for children's transitory
'regress' to counting-all. Journal for Research in Mathematics Education, 42(5), 418-450.
Tzur, R., & Simon, M. A. (2004). Distinguishing two stages of mathematics conceptual learning.
International Journal of Science and Mathematics Education, 2, 287-304.
Tzur, R., Xin, Y. P., Si, L., Kenney, R., & Guebert, A. (2010). Students with learning disability in math
are left behind in multiplicative reasoning? Number as abstract composite unit is a likely
"culprit". Paper presented at the American Educational Research Association. Retrieved from
http://www.eric.ed.gov/
95
Tzur, R., Xin, Y. P., Si, L., Woodward, J., & Jin, X. (2009). Promoting transition from participatory to
anticipatory stage: Chad's case of multiplicative mixed-unit coordination (MMUC). In M.
Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the
International Group for the Psychology of Mathematics Education (Vol. 5, pp. 249-256).
Thessaloniki, Greece: PME.
Tziir, R. (Personal Communication, November 15, 2012)
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, D.C.:
Palmer.
Wilkins, J. L., & Norton, A. (201 1). The splitting loope. Journal for Research in Mathematics Education,
42(4), 386-416.
Zazkis, R., & Hazzan, O. (1998). Interviewing in mathematics education research: Choosing the
questions. The Journal of Mathematical Behavior, 77(4), 429-439.
Zhou, Z., Peverly, S. T., & Lin, J. (2005). Understanding early mathematical competencies in American
and Chinese children. School Psychology International, 26(4), 413-427.
96
APPENDIX A
Assessment for Multiplicative Reasoning Feedback
Dear Expert/Teacher:
This assessment will be taken by students to determine their level of multiplicative
reasoning.
Please provide me with your feedback regarding the tasks.
Validity - Will this task allow me to tell if a student can reason in this way?
Grade level appropriateness -Are the problem settings and numbers used appropriate for
the grade level indicated?
Wording (Comments) - Please suggest changes that will make the problem clearer for
students.
Keep/change/omit : Indicate if item can/should be kept in the assessment as is (1),
maintained with some changes (2 - add changes in the Comments column), or omitted (3
- provide reason).
Task
Validity?
Y/N
Grade Level
Appropriate?
Y/N
Comments, Wording Suggestions
Keep/Change/Omit
Counting
On
Missing
Second
Addend
Missing
First
Addend
Composite
Unit 1
Composite
Unit 2
mDC 1
mDC2
mDC 3
97